Numerical quantum dynamics

NUMERICAL QUANTUM DYNAMICS

Progress in Theoretical Chemistry and Physics VOLUME 9

Honorary Editors: W.N. Lipscomb (Harvard University, Cambridge, MA, U.S.A.) I. Prigogine (Université Libre de Bruxelles, Belgium) Editors-in-Chief: J. Maruani (Laboratoire de Chimie Physique, Paris, France) S. Wilson (Rutherford Appleton Laboratory, Oxfordshire, United Kingdom)

Editorial Board: H. Ågren (Royal Institute of Technology, Stockholm, Sweden) D. Avnir (Hebrew University of Jerusalem, Israel) J. Cioslowski (Florida State University, Tallahassee, FL, U.S.A.) R. Daudel (European Academy of Sciences, Paris, France) E.K.U. Gross (Universität Würzburg Am Hubland, Germany) W.F. van Gunsteren (ETH-Zentrum, Zürich, Switzerland) K. Hirao (University of Tokyo, Japan) (Komensky University, Bratislava, Slovakia) M.P. Levy (Tulane University, New Orleans, LA, U.S.A.) G.L. Malli (Simon Fraser University, Burnaby, BC, Canada) R. McWeeny (Università di Pisa, Italy) P.G. Mezey (University of Saskatchewan, Saskatoon, SK, Canada) M.A.C. Nascimento (Instituto de Quimica, Rio de Janeiro, Brazil) J. Rychlewski (Polish Academy of Sciences, Poznan, Poland) S.D. Schwartz (Yeshiva University, Bronx, NY, U.S.A.) Y.G. Smeyers (Instituto de Estructura de la Materia, Madrid, Spain) S. Suhai (Cancer Research Center, Heidelberg, Germany) O. Tapia (Uppsala University, Sweden) P.R. Taylor (University of California, La Jolla, CA, U.S.A.) R.G. Woolley (Nottingham Trent University, United Kingdom)

Numerical Quantum Dynamics by

Wolfgang Schweizer Departments of Theoretical Astrophysics and Computational Physics, University Tübingen, Germany

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Progress in Theoretical Chemistry and Physics A series reporting advances in theoretical molecular and material sciences, including theoretical, mathematical and computational chemistry, physical chemistry and chemical physics

Aim and Scope Science progresses by a symbiotic interaction between theory and experiment: theory is used to interpret experimental results and may suggest new experiments; experiment helps to test theoretical predictions and may lead to improved theories. Theoretical Chemistry (including Physical Chemistry and Chemical Physics) provides the conceptual and technical background and apparatus for the rationalisation of phenomena in the chemical sciences. It is, therefore, a wide ranging subject, reflecting the diversity of molecular and related species and processes arising in chemical systems. The book series Progress in Theoretical Chemistry and Physics aims to report advances in methods and applications in this extended domain. It will comprise monographs as well as collections of papers on particular themes, which may arise from proceedings of symposia or invited papers on specific topics as well as initiatives from authors or translations. The basic theories of physics – classical mechanics and electromagnetism, relativity theory, quantum mechanics, statistical mechanics, quantum electrodynamics – support the theoretical apparatus which is used in molecular sciences. Quantum mechanics plays a particular role in theoretical chemistry, providing the basis for the valence theories which allow to interpret the structure of molecules and for the spectroscopic models employed in the determination of structural information from spectral patterns. Indeed, Quantum Chemistry often appears synonymous with Theoretical Chemistry: it will, therefore, constitute a major part of this book series. However, the scope of the series will also include other areas of theoretical chemistry, such as mathematical chemistry (which involves the use of algebra and topology in the analysis of molecular structures and reactions); molecular mechanics, molecular dynamics and chemical thermodynamics, which play an important role in rationalizing the geometric and electronic structures of molecular assemblies and polymers, clusters and crystals; surface, interface, solvent and solid-state effects; excited-state dynamics, reactive collisions, and chemical reactions. Recent decades have seen the emergence of a novel approach to scientific research, based on the exploitation of fast electronic digital computers. Computation provides a method of investigation which transcends the traditional division between theory and experiment. Computer-assisted simulation and design may afford a solution to complex problems which would otherwise be intractable to theoretical analysis, and may also provide a viable alternative to difficult or costly laboratory experiments. Though stemming from Theoretical Chemistry, Computational Chemistry is a field of research

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Progress in Theoretical Chemistry and Physics in its own right, which can help to test theoretical predictions and may also suggest improved theories. The field of theoretical molecular sciences ranges from fundamental physical questions relevant to the molecular concept, through the statics and dynamics of isolated molecules, aggregates and materials, molecular properties and interactions, and the role of molecules in the biological sciences. Therefore, it involves the physical basis for geometric and electronic structure, states of aggregation, physical and chemical transformations, thermodynamic and kinetic properties, as well as unusual properties such as extreme flexibility or strong relativistic or quantum-field effects, extreme conditions such as intense radiation fields or interaction with the continuum, and the specificity of biochemical reactions. Theoretical chemistry has an applied branch – a part of molecular engineering, which involves the investigation of structure–property relationships aiming at the design, synthesis and application of molecules and materials endowed with specific functions, now in demand in such areas as molecular electronics, drug design or genetic engineering. Relevant properties include conductivity (normal, semi- and supra-), magnetism (ferro- or ferri-), optoelectronic effects (involving nonlinear response), photochromism and photoreactivity, radiation and thermal resistance, molecular recognition and information processing, and biological and pharmaceutical activities, as well as properties favouring self-assembling mechanisms and combination properties needed in multifunctional systems. Progress in Theoretical Chemistry and Physics is made at different rates in these various research fields. The aim of this book series is to provide timely and in-depth coverage of selected topics and broad-ranging yet detailed analysis of contemporary theories and their applications. The series will be of primary interest to those whose research is directly concerned with the development and application of theoretical approaches in the chemical sciences. It will provide up-to-date reports on theoretical methods for the chemist, thermodynamician or spectroscopist, the atomic, molecular or cluster physicist, and the biochemist or molecular biologist who wish to employ techniques developed in theoretical, mathematical or computational chemistry in their research programmes. It is also intended to provide the graduate student with a readily accessible documentation on various branches of theoretical chemistry, physical chemistry and chemical physics.

vi

Contents

List of Figures List of Tables Preface

ix xv xvi

1. INTRODUCTION TO QUANTUM DYNAMICS 1. The Schrödinger Equation 2. Dirac Description of Quantum States Angular Momentum 3. The Motion of Wave Packets 4. The Quantum-Classical Correspondence 5.

1 1 7 9 14 19

2. SEPARABILITY 1. Classical and Quantum Integrability Separability in Three Dimensions 2. Coordinates and Singularities 3.

35 35 38 49

3. APPROXIMATION BY PERTURBATION 1. The Rayleigh-Schrödinger Perturbation Theory l/N-Shift Expansions 2. Approximative Symmetry 3. Time-Dependent Perturbation Theory 4.

55 56 70 77 85

4. APPROXIMATION TECHNIQUES 1. The Variational Principle 2. The Hartree-Fock Method Density Functional Theory 3. 4. The Virial Theorem 5. Quantum Monte Carlo Methods

95 95 104 106 108 118

5. FINITE DIFFERENCES 1. Initial Value Problems for Ordinary Differential Equations 2. The Runge-Kutta Method vii

133 133 135

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Predictor-Corrector Methods Finite Differences in Space and Time The Numerov Method

139 140 150

6. DISCRETE VARIABLE METHOD 1. Basic Idea Theory 2. Orthogonal Polynomials and Special Functions 3. 4. Examples The Laguerre Mesh 5.

155 155 157 165 182 200

7. FINITE ELEMENTS Introduction 1. Unidimensional Finite Elements 2. Adaptive Methods: Some Remarks 3. 4. B-Splines Two-Dimensional Finite Elements 5. Using Different Numerical Techniques in Combination 6.

209 210 212 230 232 235 248

8. SOFTWARE SOURCES Acknowledgments

255 261

3. 4. 5.

Index 263

List of Figures

1.1

1.2 2.1

2.2

Graphic example for an active coordinate rotation. On the lefthand side top, the body fixed and laboratory coordinate system are equal. To follow the coordinate rotation, the sphere is covered with a pattern. On the lefthand side of the sphere there is a white strip running from north pole to south pole. The pole axis is the Z-axis. First step: Rotation around the Z-axis. This gives the sphere on the top righthand side. Second step: rotation around the new which leads to the sphere on the bottom lefthand side, and last step, rotation around the new Potential well and the classical turning points defined by Energy in Rydberg for the 750th to 759th positive parity diamagnetic Rydberg states in dependence of the magnetic field strength measured in units of (From [7]) Histogram of the nearest neighbor distribution for the diamagnetic hydrogen atom. For the computation about 350 positive parity, Rydberg eigenstates have been used. On the lefthand side for and on the righthand side for with the magnetic field strength measured in units of and E the energy in Hartrees. For the lefthand side the classical corresponding system is almost regular and for the righthand side almost completely chaotic. For more details see [9]. For comparison the Poisson and the Wigner distribution are additionally plotted. ix

12 30

37

38

x

NUMERICAL QUANTUM DYNAMICS

2.3

3.1

3.2

3.3

4.1

5.1

The nuclei are situated at points “P1” and “P2”, the electron at “e”. R is the distance between the two nuclei, and r1 respectively r2 the distances of the electron to each of the nuclei. The Cartesian coordinate system is chosen such that is the z-axis with the origin in the middle of

46

On top of the figure we show the energy as a function of the magnetic field strength for the states (left, top) and The prime indicates that this labeling would only be correct in the field free limit. At a certain magnetic field strength the two lines cross each other – the two states are degenerate. At the bottom we show the corresponding probability of presence in cylindrical coordinates. The magnetic field B points into the zdirection. The wave functions remain undistorted in the region where the two lines cross each other. (From [5])

63

The states are the same as in Fig. (3.1) and their corresponding energies are plotted in dependence of the magnetic field strength, but now with an additional parallel electric field of magnitude The parity is no longer a good quantum number and hence degeneracy becomes forbidden. Therefore, we observe an avoided crossing: the states interact with each other and the wave functions are distorted clearly close to the point of the forbidden crossing. (From [5])

64

Transition probability in units of tion of

88

as a func-

Coulomb (C), exchange (A) and overlap (S) integral as a function of the nuclei distance in atomic units.

117

Diagrammatic picture of the time propagation around a pulse. 141

List of Figures

5.2

5.3 5.4

6.1 6.2

xi

In all rows the hight of the initial Gaussian wave packet and of the potential wall is selected such, that it equals the relative ratio between the kinetic and the potential energy. On the left hand side tunneling occurs. In the middle the kinetic energy of the initial wave packet is higher than the potential energy, and on the left hand side we have the same situation but now with two Gaussians traveling in opposite directions. From top to bottom: the propagated wave packet is shown in equal time steps of T/6. (The numerical values are selected as described in Chapt. 5.4.3.1.)

145

Potential and the lowest five corresponding eigenfunctions, labeled by to

148

Ground state and first excited state of the harmonic oscillator. On the lefthand side computed with the correct initial conditions, on the righthand side the erroneous computation with the opposite parity.

152

On top the Legendre polynomials (solid lines), and bottom and

and (solid lines).

173

Energy of the 3rd eigenstate for the quartic anharmonic oscillator as a function of ‘pert.’ are the results in 1st order perturbation theory. The 3-dimensional DVR positive parity computations are the bold solid line and the diamonds indicate the 6-dimensional computations. The thin solid line is obtained by diagonalizing the 6dimensional positive parity Hamiltonian matrix.

185

6.3

Eigenenergies and potential curve for the general anharmonic oscillator. Superimposed (dots) are the analytic results of the harmonic oscillator for comparison. On the lefthand side for and on the righthand side From top to bottom: in steps of 0.05. 186

6.4

Eigenfunctions of the discrete variable representation for N = 3. On top for the fixed-node and on bottom for the periodic DVR. The nodes for the fixed-node representation are and for the periodic representation obviously the Kronecker delta property is satisfied.

197

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NUMERICAL QUANTUM DYNAMICS

7.1

7.2

7.3

Top: Graphical example for a radial hydrogen wave function. The entire space is divided into small elements. On each of the elements the wave function is expanded via interpolation polynomials. Here each element has 5 nodal points at 0, 0.25, 0.5, 0.75 and 1. Bottom: For Coulomb systems convergence is significantly improved by quadratically spacing the elements. This means that the size of the elements increases linearly and the distance from the origin increases quadratically with the number of elements n.

214

Top: Graphical example for the nodes of the lowest 10 excited radial hydrogen wave functions for l = 0, Bottom: The nodes of the first 10 oscillator eigenstates. The distance between neighboring nodes of a radial hydrogen wave function increases significantly as a function of the distance from the origin, whereas the distances between neighboring nodes of the oscillator eigenstates are always of the same order.

216

The lowest four Lagrange interpolation polynomials. The nodes are equidistant in [–1,1]. Thus for two nodes, (–1, +1), the interpolation polynomials are linear; for three nodes, (–1, 0,+1) quadratic and so on. Obviously each of the interpolation polynomials fulfill 219

7.4

7.5

7.6

Hermite interpolation polynomials. Left hand side for two and right hand side for three nodes. Top the interpolation polynomials, bottom the derivative of the interpolation polynomials. "1" and "2" on the left hand side and "1", "2" and "3" on the right hand side label the polynomials the others label the Hermite interpolation polynomials see Eq.(7.35a).

222

Extended Hermite interpolation polynomials for two nodes. Left hand side the interpolation polynomials, center its first and right hand side its second derivative. "1" and "2" label the polynomials , "3" and "4" and "5" and "6"

224

Example for the global Hamiltonian matrix for 5 finite elements. Each of the blocks are given by the local matrix.

226

List of Figures

7.7

7.8

7.9 7.10 7.11 7.12 7.13 7.14

7.15

7.16

7.17

Convergence of the energy for the state as a function of the number of non-vanishing Hamiltonian matrix elements Top, "1", for linear Lagrange interpolation polynomials, "3" for third order and "5" for 5-th order interpolation polynomial. Thick lines for Lagrange and thin lines for Hermite interpolation polynomials. Convergence for the state in dependence of the element structure for Hermite interpolation polynomials. and is the number of non-vanishing Hamiltonian matrix elements. "const." for finite elements of constant size, "quadr." for quadratically spaced finite elements, and "lag." for finite elements whose dimension is given by the zeros of a Laguerre polynomial. Triangle in global coordinates. Triangular element in local coordinates. Grid and grid labels for triangular finite elements. Linear interpolation polynomial Quadratic interpolation polynomials. (Lefthand side and righthand side Lefthand side finite element for quadratic and righthand side for cubic two-dimensional interpolation polynomials. Cubic interpolation polynomials. From lefthand side top to righthand side bottom: interpolation polynomial and Interpolation polynomials for finite elements with 15 nodes. Top: For finite elements with nodes only on the border; bottom for finite elements with nodes on the border and three internal nodes. Lefthand interpolation polynomial righthand side interpolation polynomial Complex energy of the hydrogen ground state for F=0.3 as a function of the complex rotation angle To uncover the “convergence trajectory” the deviation from the converged value is plotted: and with the real part of the complex energy eigenvalue and the FWHM of the resonance.

xiii

228

229 236 237 238 239 239

241

242

243

251

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List of Tables

3

1.1

Some partial differential equations in physics.

3.1

Contribution in % of the 0th to 4th perturbational orders to the final energy for different nuclei, for the ground "0" and the first excited "1" breathing modes.

77

Eigenvalues of the diamagnetic contribution for some diamagnetic hydrogenic eigenstates. indicates the field-free quantum number and the upper index which states form a common adiabatic multiplet. States without upper index are adiabatic singlet states.

84

Comparison of the energy of the state as function of various external field strengths obtained by the approximate invariant (inv.) and by ‘exact’ numerical computations via discrete variable and finite elements.

85

3.2

3.3

4.1

Ionization potential for two-electron systems in Rydberg

100

5.1

The lowest five eigenenergies for the Pöschl-Teller potential in dependence of the number of finite difference steps.

149

The lowest five eigenenergies for the harmonic oscillator. denotes the integration border and dx the step size.

150

5.2

5.3

The lowest five eigenenergies for the anharmonic oscillator with potential denotes the left and the right integration border and dx the step size. xv

150

xvi

NUMERICAL QUANTUM DYNAMICS

6.1

6.2

7.1 7.2

Parameters for the model potential is the averaged relative error over all computed eigenstates up to principal quantum number n= 10 and the difference between our computation and the experimental result. Energy in Rydberg for the ground state of the hydrogen atom for spin down states for The Laguerre mesh was obtained by with respectively Some polynomial coefficients for the Lagrange interpolation polynomials as defined by Eq.(7.30) Eigenvalues for the two-dimensional harmonic oscillator. For both computations quadratic interpolation polynomials are used.

188

203 220

247

Preface

It is an indisputable fact that computational physics form part of the essential landscape of physical science and physical education. When writing such a book, one is faced with numerous decisions, e.g.: Which topics should be included? What should be assumed about the readers’ prior knowledge? How should balance be achieved between numerical theory and physical application? This book is not elementary. The reader should have a background in quantum physics and computing. On the other way the topics discussed are not addressed to the specialist. This work bridges hopefully the gap between advanced students, graduates and researchers looking for computational ideas beyond their fence and the specialist working on a special topic. Many important topics and applications are not considered in this book. The selection is of course a personal one and by no way exhaustive and the material presented obviously reflects my own interest. What is Computational Physics? During the past two decades computational physics became the third fundamental physical discipline. Like the ‘traditional partners’ experimental physics and theoretical physics, computational physics is not restricted to a special area, e.g., atomic physics or solid state physics. Computational physics is a methodical ansatz useful in all subareas and not necessarily restricted to physics. Of course this methods are related to computational aspects, which means numerical and algebraic methods, but also the interpretation and visualization of huge amounts of data. Similar to theoretical physics, computational physics studies the properties of mathematical models of the nature. In contrast to analytical results obtained by theoretical models, computational models can be much closer to the real world. Due to the numerical backbone of computational physics more details can be studied. Experimental physics is an empirical ansatz to obtain a better xvii

xviii

NUMERICAL QUANTUM DYNAMICS

understanding of the world. Computational physics has as well a strong empirical component. For fixed parameters of our computational model of the real-world physical system we obtain a single and isolated answer. Parameter studies result in a bundle of numbers. Intuition and interpretation is necessary to obtain a better understanding of nature from these numbers - the number by its own, without any interpretation, is nothing. Mathematical models describing classical systems are dominated by ordinary differential equations, nonrelativistic quantum dynamics by the Schrödinger equation, which leads for bound states to an elliptic boundary value problem. The eigenfunctions of the Schrödinger equation for bound states will vanish in the infinity. Thus the computational methods useful in solving physical models will also differ between classical mechanics, electrodynamics, statistical mechanics or quantum mechanics and so forth - although there will be a certain overlap. Methods useful for a small number of degrees-of-freedom cannot be used for many particle systems. Even if the computer power - storage and speed - increased dramatically during the last decades, many body systems cannot be treated on the same computational footing as, e.g., two degreesof-freedom systems. In this book we will concentrate on numerical methods adequate mainly for small quantum systems. The book consists of eight chapters. Chapter one formalizes some of the ideas of quantum mechanics and introduces the notation used throughout the book. The Dirac description will be briefly discussed, angular momentum, the Euler angles and the Wigner rotation function introduced. The motion of wave packets and some comments about the discretization of the time propagator can be also found there. This chapter will be completed by a discussion of the quantum classical correspondence, including the WKB approximation and the representation of quantum wave functions in phase space. Chapter two, devoted to the discussion of integrability and separability, contains a list of all coordinate systems for which the three-dimensional Schrödinger equation could be separable. Chapter three and four revisits approximation techniques. We will start with Schrödinger perturbation theory and discuss briefly the effect and the qualitative consequences of perturbing degenerate states. Approximative symmetries and dynamical Lie algebras will be as well a topic. Advanced computational methods are often justified by variational principles. Thus we will discuss the Rayleigh-Ritz variational principle. Many numerical techniques will lead to banded or sparse Hamiltonian matrices. Effective linear algebra routines are based on Krylov space techniques like the Arnoldi and Lanczos method, which will be introduced in this chapter. The many-body Schrödinger equation cannot be solved directly. Thus approximation models have to be used to obtain sensitive results. Hartree-Fock and density functional theory are two of the most

Preface

xix

important techniques to deal with many body problems. We will briefly discuss these two methods, followed by a discussion of the virial theorem. The last section of chapter four deals with quantum Monte Carlo methods, which due to the increasing computer power will become more important for quantum computations in future. Chapter five contains finite differences. Here we will also discuss initial value problems for ordinary differential equations. Central in this chapter is the discretization of the uni-dimensional time-dependent Schrödinger equation in the space and time variable. This chapter will be completed by deriving the Numerov method, useful for computing bound states, resonances and scattering states. The theory of discrete variables is the subject of chapter six. The essential mathematical background is provided. Orthogonal polynomials are discussed and the computation of their nodes are explained. These nodes play an important rôle in applying the discrete variable technique to quantum systems. Some examples are discussed emphasizing the combination of discrete variable techniques with other numerical methods. Chapter seven is devoted to the discussion of finite elements. We will discuss one- and two-dimensional finite elements, the discussion of two-dimensional elements will be restricted to elements with triangular shape. We will derive interpolation polynomials useful for quantum applications and introduce Spline finite elements and discuss, by an example, the combination of finite elements for the radial coordinate with discrete variables in the angular coordinate. Many computational techniques necessitate linear algebra routines to obtain the results. Adaptive techniques for the grid generation are useful to optimize finite element computations. Other routines are necessary to compute the nodes of orthogonal polynomials, fast Fourier transformations to obtain wave functions in phase space, graphical routines to visualize numerical results and so on. This list could become endless easily. For the benefit of the reader the final chapter contains a - of course incomplete - list of useful sources for software.

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Chapter 1 INTRODUCTION TO QUANTUM DYNAMICS

This chapter gives a brief review of quantum mechanics. Although the reader is expected to have some experience in the subject already, the presentation starts at the beginning and is self-contained. A more thorough introduction to quantum mechanics can be found in numerous monographs, e.g., [1, 2, 3]. In the present state of scientific knowledge the importance of quantum mechanics is commonplace. Quantum mechanics play a fundamental rôle in the description and understanding of physical, chemical and biological phenomena. In 1900, Max Planck [4] was the first to realize that quantum theory is necessary for a correct description of the radiation of incandescent matter. With this insight he had opened new horizons in science. Two further historic cornerstones are Einstein’s photoelectric effect and 1923 de Broglie’s – at that time – speculative wave-particle duality. In accordance with Einstein’s insight and de Broglie’s hypothesis Schrödinger1 derived in 1925 his famous Schrödinger equation.

1.

THE SCHRÖDINGER EQUATION

This section is an introduction to the Schrödinger equation. No attempt is made here to be complete. The essential goal is to lay the basis for the rest of the book – the numerical aspects of non-relativistic quantum theory.

1.1

INTRODUCTION

The emission of electrons from the surface of a metal was discovered by Hertz in 1887. Later experiments by Lenard showed that the kinetic energy of the emitted electron was independent of the intensity of the incident light and that there was no emission unless the frequency of the light was greater than a threshold value typical for each element. Einstein realized that this is what 1

2

NUMERICAL QUANTUM DYNAMICS

is to be expected on the hypothesis that light is absorbed in quanta of amount In the photoelectric effect an electron at the surface of the metal gains an energy by the absorption of a photon. The maximum kinetic energy of the ejected electron is where is the contact potential or work function of the surface. Einstein’s theory predicts that the maximum kinetic energy of the emitted photoelectron is a linear function of the frequency of the incident light, a result which was later confirmed experimentally by Millikan (1916) and which allowed to measure the value of the Planck constant These results lead to the conclusion, that the interaction of an electromagnetic wave with matter occurs by means of particles, the photons. Corpuscular parameters (energy E and momentum of the photons are related to wave parameters (frequency and wave vector of the electromagnetic field by the fundamental Planck-Einstein relations:

where with Js the Planck constant. Newton was the first to resolve white light into separate colors by dispersion with a prism. The study of atomic emission and absorption spectra uncovered that these spectra are composed of discrete lines. Bohr assumed 1913 that an electron in an atom moved in an orbit around the nucleus under the influence of the electrostatic attraction of the nucleus just like the planets in the solar system. Due to classical electrodynamics an accelerated electron will radiate and lose energy. Thus such a classical system would be unstable. Therefore Niels Bohr assumed that the electron moves on quantized orbits and that the electron radiate only when they jump from an higher to a lower orbit. Of course the origin of this quantization rules, the Bohr-Sommerfeld quantization, remained mysterious. In 1923 de Broglie formulated the hypothesis, that material particles, just like photons, can have wavelike aspects. Hence he associated with particle parameters energy and momentum wave frequencies and wave vectors, similar to the Planck-Einstein relation (1.1). Therefore the corresponding wavelength of a particle is

By this relation de Broglie could derive the Bohr-Sommerfeld quantization rule. At that time Erwin Schrödinger was working at the institute of Peter Debye in Zürich, and Peter Debye suggested to derive a wave equation for these strange de-Broglie-waves, which resulted in the famous time-dependent Schrödinger

Introduction to Quantum Dynamics

3

equation

By acting with the potential-free Schrödinger equation on a plane de-Brogliewave he obtained the correct de Broglie relation (1.2) and he could derive the hydrogen eigenspectrum2 .

Mathematical aspects: partial differential equations. The mathematical typology of quasilinear partial differential equations of 2nd order

distinguishes three types of such equations: hyperbolic, parabolic or elliptic. A quasilinear partial differential equation of 2nd order, Eq.(1.4), is called

with In physical applications hyperbolic and parabolic partial differential equations usually describe initial value problems and elliptic equations boundary value problems. In Tab. (1.1) some typical physical examples are listed.

Physical aspects. The solutions of the Schrödinger equation are complex functions. Due to the interpretation of Max Born, the modulus square of the normalized wave function at a position gives us the probability

4

NUMERICAL QUANTUM DYNAMICS

of finding the particle at that point. Heisenberg discovered that the product of position and momentum uncertainties and is greater than or equal to Planck’s constant

This uncertainty can be generalized to arbitrary operators. The expectation value of an operator  is defined by

and its variance by

in dimensions). A vanishing variance of an Hermitian operator means that its expectation value can be measured exactly. The commutator of two operators  , is defined by

If  and are two Hermitian operators which do not commute, they cannot be both sharply measured simultaneously. Let

then the uncertainties

and

hold

Coming back to the probability interpretation, we must require that

since at some initial

the probability density and

the particle must be somewhere in space. Let

Introduction to Quantum Dynamics

5

the probability current density or flux, then we get the following law of probability conservation

This is a continuity equation analogous to the one between the charge and current densities in electrodynamics and insures that the physical requirement of probability conservation is satisfied. The momentum operator has the coordinate-space representation and thus the differential operator in the probability current density can be interpreted as the velocity operator of the quantum particle under consideration. Therefore if the particle is placed in an electromagnetic field the momentum operator has to be replaced by with q the particle’s charge and the vector potential of the external electromagnetic field.

1.2

THE SCHRÖDINGER EQUATION AS AN EIGENVALUE PROBLEM: HAMILTONIAN OPERATOR

Let us return to the time-dependent Schrödinger equation (1.3) under the assumption of a time independent potential By the standard mathematical procedure we can separate the time and space variables in the time-dependent Schrödinger equation (1.3). This gives

and because only the lefthand side of the differential equation depends on the time variable t, respectively the righthand side only on space variables, this equation can be satisfied only if both sides are equal to a constant. Calling this constant E, we obtain

and the time-independent Schrödinger equation

Consequently this can also be written as an eigenvalue equation

6

NUMERICAL QUANTUM DYNAMICS

with the Hamiltonian, the eigenfunction of the Hamilton operator and the constant E its eigenvalue. Thus whereas the time-dependent Schrbödinger equation describes the development of the system in time, the time independent equation is an eigenvalue equation and its (real) eigenvalue the total energy of the system. This set of eigenvalues could be discrete with normalizable eigenvectors, or a continuous one, or a mixed one as in the case of the hydrogen atom. A mixed spectrum may have a finite or even infinite number of discrete eigenvalues. Bound states are always discrete. Thus the question occurs under which condition an attractive potential possesses bound states. In one-dimensional potentials it is a well-known fact that a bound state exists if for all x or even if satisfies the weaker condition More of interest are attractive, regular 3-dimensional central potentials. Regular potentials are those which are less singular than at the origin, and go to zero faster than at infinity, r being the radial coordinate. More precisely, they satisfy the condition

For those potentials Bargmann has proved the inequality

where is the number of bound states with angular momentum l, Calogero [6] have shown that the number of bound states for the radial Schrödinger equation in the S-wave (l = 0) admits the upper bound

These upper bounds have been recently generalized[7] for central regular potentials with to

(For more details and additional inequalities for upper bounds see [7].) Resonances. Strictly speaking, the continuous spectrum of an Hamiltonian has no eigenvector, because the corresponding state is no longer normalizable. Any real quantum system has quasibound states or resonances which are important for understanding their dynamics. Like the continuum and scattering states resonances are associated with non-normalized solutions of the timeindependent Schrödinger equation. However, the complex coordinate method

Introduction to Quantum Dynamics

7

or complex coordinate rotation [8] enables us to isolate the resonance states from the other states and to discretize the continuum. In the complex coordinate method the real configuration space coordinates are transformed by a complex dilatation or rotation. The Hamiltonian of the system is thus continued into the complex plane. This has the effect that, according to the boundaries of the representation, complex resonances are uncovered with square-integrable wavefunctions and hence the space boundary conditions remain simple. This square integrability is achieved through an additional exponentially decreasing term

After the coordinates entering the Hamiltonian have been transformed, the Hamiltonian is no longer hermitian and thus can support complex eigenenergies associated with decaying states. Thereby a complex Schrödinger eigenvalue equation is obtained. The spectrum of a complex-rotated Hamiltonian has the following features [8]: Its bound spectrum remains unchanged, but continuous spectra are rotated about their thresholds into the complex plane by an angle of Resonances are uncovered by the rotated continuum spectra with complex eigenvalues and square-integrable (complex rotated) eigenfunctions. The complex eigenvalue yields the resonance position and width The inverse of the width gives the lifetime of the state. This can be understood easily by the separation ansatz Eq.(1.12). Due to Eq.(1.14) we obtain for an eigenstate of the complex coordinate rotated Hamiltonian

and thus this state will exponentially decay. The complex coordinate method has been applied to various physical phenomena and systems. For a recent review and more details see [9], In Eq.(l .6) we have already used Dirac’s “bras” and “kets” as an abbreviation for computing the expectation of an operator. In the following section we will briefly discuss Dirac’s bra-ket notation, which has the advantage of great convenience and which uncovers more clearly the mathematical formulation of quantum mechanics.

2.

DIRAC DESCRIPTION OF QUANTUM STATES

This section is an introduction to the Dirac notation, to the “bra” and “kets”, which will be mainly used throughout the book. For more details see Dirac’s famous book [2]. Observables of quantum systems are represented by Hermitian operators Ô in Hilbert space, the states by generalized vectors, the state vector or wave

8

NUMERICAL QUANTUM DYNAMICS

function. Following Dirac we call such a vector ket and denote it by This ket is postulated to contain complete information about the physical state. The eigenstates of an observable Ô are specified by the eigenvalues and the corresponding eigenvalue equation is

with set, so

the eigenstate or eigenvector. The eigenstates form an orthonormal

By this equation the bra3 vector is defined, an element of the bra space, a vector space dual to the ket space. If is an Hilbert space element is a complex number. Thus by the “bras” we have defined a map from the Hilbert space onto the complex numbers, the inner or scalar product. We postulate two fundamental properties of the inner product. First

in other words

and

are complex conjugate to each other, and second

This is also called the postulate of positive definite metric. The eigenkets of the position operator satisfying

are postulated to form a complete set and are normalized such that Hence the ket of an arbitrary physical state can be expanded in terms of

and is the coordinate representation of the state we define the momentum representation of the state thus

In analogue, and

hence is the Fourier transformation (FT) from momentum space to coordinate space.

Introduction to Quantum Dynamics

3. 3.1

9

ANGULAR MOMENTUM BASIC ASPECTS

In the early times of quantum mechanics the hydrogen spectrum was explained by quantization conditions for the electron’s motion - orbits in three dimensions. Thus it is not surprising that in quantum dynamics the concept of angular momentum plays a crucial role. The classical definition of the angular momentum is with and canonical conjugate coordinates and momenta, and we will adopt the same definition for the (orbital) angular momentum operator The components of the angular momentum operator hold the following commutator relations:

which are sometimes summarized by the easy-to-memorize expressions

What follows from the existence of the commutation relations? Due to Eq.( 1.9) we cannot simultaneously assign eigenvalues to all three components of the angular momentum operator. Thus a measurement of any component of angular momentum introduces an uncontrollable uncertainty of the order in our knowledge of the other two components. In contrast we can simultaneously measure the angular momentum square with any component of since these operators commute

Thus the angular momentum eigenstates can be labeled by the eigenvalues of and one angular momentum component. It is customary to choose and Denoting the corresponding eigenvalues by / and we obtain

Due to the angular momentum commutator relations the expectation values < Ô > vanish for the observables and for (E.g., due to Eq.(l .9) and because For practical purposes it is more comfortable to define the non-hermitian ladder operators

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which fulfill the commutator relations

and In spherical coordinates (see chapter 2.2.2) the angular momentum operators hold

and the eigenfunctions of harmonics

and

in spherical coordinates are the spherical

with are associated Legendre functions (see Chapt.5.3.2 for more details). In addition to the orbital angular momentum quantum particles possess spin, an “angular-momentum-like” operator. Since the orbital angular momentum depends only on space coordinates and spin has nothing to do with space, both operators commute. The total angular momentum is defined by with the spin operator. In general, let and be two angular momenta, and let then the total angular momentum is The corresponding eigenkets can be labeled either by or by with the quantum numbers of the single angular momenta and J, M the quantum numbers of the total angular momentum. The Clebsch-Gordan coefficients (CG) are the unitary transformation between both sets [10]

Introduction to Quantum Dynamics

The permitted values of J range from The total number of states satisfies

11

in steps of one, and for all possible J’s

The 3j-symbols are modified Clebsch-Gordan coefficients, but computationally more useful due to their symmetry properties. They are defined by

The symmetry properties of the 3j-symbols are best uncovered by the Regge symbols [11], which are defined by

The Regge symbol vanishes if one of its elements become negative or if one of its columns or rows is unequal Hence all symmetries and selection rules of the 3j-symbol are uncovered by the Regge symbol (see also Chapt. 5.4.2.2). (Unfortunately the used symbols and the exact definition of the symbols, e.g. by differs from author to author.)

3.2

COMPUTATIONAL ASPECTS

The following formulas are based on the conventions used in the book by Edmonds [12]. The Euler angles. To define the orientation of a three-dimensional object, three angles are needed. Most common are the Euler angles, which can be defined by a finite coordinate rotation between the laboratory and the body fixed frame. Transforming the coordinate system is called passive transformation, while rotating the physical object is called active transformation. In most textbooks, see e.g [12], the coordinate transformation is graphically presented. In order to appreciate the Euler angles, Fig.( 1.1) shows as an example the active view of a finite coordinate rotation defined by Euler angles. The generally accepted definition of the Euler angles is to start with the initial system X , Y, Z and a first rotation by about its Z axis. This results in a new system which is now rotated through an angle about its leading to another intermediary system: which by rotation through the third Euler angle about the passes finally into the system These

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transformations has been summarized in a short hand notation in Eq.(1.43).

Introduction to Quantum Dynamics

13

Let denote the angular momentum operator with respect to the axis a. The rotation of a wave function is given by

with

In order to perform these rotations by rotations about the original axes (X, Y, Z) we replace the rotation by first turning back to the original axis and then to rotate instead of the new axis around the original axis and going back to the new coordinate system. Thus we obtain, e.g., for

Continuing with this process will finally lead to

The Wigner Rotation Function. Let us now apply these rotations to the spherical harmonics

Because the angular momentum operator can only affect the magnetic quantum number l will be conserved. Therefore

with

where

is called the Wigner rotation function given by

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Instead of using the direct summation the Wigner rotation function can be computed alternatively via Jacobi polynomials

Jacobi polynomials are discussed in Chapt. 5.3.8 and useful recursion relations can be found there. (In symbolic computer languages like Maple the Jacobi polynomials are predefined and thus the equation above is especially useful to compute the Wigner rotation function using Maple.) 3j-symbols. By combining the definition of the 3j-symbols with the Wigner rotation functions one arrives at

where the sum runs over values of s for which the argument of the factorial is nonnegative. (Additional recursion relations can be found in [12].)

4. 4.1

THE MOTION OF WAVE PACKETS THE TIME PROPAGATOR

In contrast to coordinates and momenta, time is just a parameter in quantum mechanics. Thus time is not connected with an observable or an operator. Our basic concern in this section is the motion of a wave packet or ket state. Suppose we have a physical system prepared at time At later times, we do not expect the system to remain in the initially prepared state. The time evolution of the state is governed by the Schrödinger equation. Because the Schrödinger equation is a linear equation, there exists a linear (unitary) operator, which maps the initial wave packet onto the wave packet at a later time t

Because of

and due to

Introduction to Quantum Dynamics

the time evolution operator or time propagator

15

is transitive

and because of U hold From the time-dependent Schrödinger equation we get immediately

and thus the integral equation

Conservative quantum systems. For a conservative quantum system the Hamiltonian is time independent. The solution to Eq.(1.51) in such a case is given by

To evaluate the effect of the time propagator on a general initial wave packet, we can either expand the wave packet with respect of the Hamiltonian eigenfunctions or expand the time evolution operator in terms of the eigenprojector. Because both ways are educational we will discuss both briefly - of course the results are entirely equivalent. Thus let us start with the complete eigenbasis of the Hamiltonian: and expand our wave packet with respect to the eigenbasis

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because Equivalent to the method above is the direct expansion of the time evolution operator by the idempotent projector

and thus again

Of course this equation could also be derived directly using the time-dependent Schrödinger equation. Non-conservative systems. For non-conservative systems is explicitly time dependent and thus the integral equation (1.51) cannot be solved directly. Nevertheless there is a simplification for Hamiltonians which commute at different times, which is rather an exception than the rule. The formal propagator for those systems can be computed by

To derive a formal solution in the general case let us iteratively interpret the integral equation (1.51). Hence

and this series is sometimes called Neumann series or Dyson series, after F. J. Dyson, who developed a similar perturbation expansion for the Green function

Introduction to Quantum Dynamics

in quantum field theory. Note, that because the time order becomes important.

4.2

17

does not commute with

DISCRETIZATION IN TIME

4.2.1 THE CAYLEY METHOD Let us restrict the following discussion to conservative systems. Hence the time evolution of a wave packet could be described by Eq.(1.52). Suppose we know a sufficiently large subset of the eigenstates of the systems, then we could expand our initial wave packet with respect of these eigenfunctions. Such a formal expansion could cause several numerical problems: In case we know the eigensolutions only approximately and/or the summation converges only weakly, computational errors will accumulate by adding further and further contributions to Eq.(l .55). In case of a mixed spectrum not only the bound states but in addition the continuum states have to be considered, and of course even the computation of the eigenstates could be hard work. Therefore especially if the continuum becomes important, e.g via tunneling, a direct approximate solution of seems more favorable. The naïve approximation by a first order Taylor expansion

fails because the approximating operator is non-unitary and hence the norm of the wave packet will not be conserved. This problem could be overcome easily and additionally the accuracy improved. Let us again start with the correct propagation.

and this approximation is called Cayley or Crank-Nicholson approximation. A closer look reveals that this formula is now of second order in and that the approximative propagator is again unitary, because

A didactical very nicely written paper, in which time dependent aspects of tunelling and reflexion of wave packets for one-dimensional barrier potentials

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are discussed, can be found in [13]. These authors combined a finite difference method in the space coordinate with the Cayley method for the time variable (for more details see chapter 4.4). For one-dimensional systems the implicit equations resulting from this discretization procedure can be solved iteratively. For higher dimensions this would be no longer possible and thus a coupled system of linear equations have to be solved. An application can be found in [14]. 4.2.2 THE PEACEMAN-RACHFORD METHOD The essential idea of the Peaceman-Rachford computational scheme is to split the propagator in two parts. Let  and be two operators that do not necessarily commute, but whose commutator fulfills

Then

with x a complex number. This is called the Campbell-Baker-Hausdorff theorem. Thus for commuting operators the operator sum in the exponential function can be mapped onto the operator product of the exponential operators. Therefore if the Hamiltonian can be written in a sum it could be numerical of interest first to separate the propagator into two sub-propagators and to expand each of this sub-propagators in a Cayley-like expansion. For sufficiently small time steps and the Hamiltonian we obtain

and with Eq.(1.58)

thus

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19

Computationally it is more efficient to separate the action of this approximate time propagator on the wave packet into two implicit schemes by introducing an artificial intermediate wave packet

It is very common in literature to denote this intermediate state by But note, this notation is misleading because this is not a propagated wave packet, it is only a “book keeping” time-label and numerically an intermediate step towards the propagated wave packet at time Thus numerically we solve first the Eq.(1.65a) and then substitute this result in Eq.(1.65b). Like the Cayley method this propagation scheme preserves unitarity and is for commuting operators of second order in t, but for non-commuting operators only of first order.

5.

THE QUANTUM-CLASSICAL CORRESPONDENCE

In the early 20th century physicists realized that describing atoms in purely classical terms is doomed to failure. Ever since the connection between classical mechanics and quantum mechanics has interested researchers. The rich structure observed in low-dimensional non-integrable classical systems has renewed and strengthen this interest [15]. For atoms, it was always expected that for energy regions in which the energy spacing between neighboring states is so small as to resemble the continuum as predicted by classical mechanics, quantum effects become small. Thus todays textbooks state that the classical limit will be reached for highly excited states. That this is an oversimplification was shown by Gao [16]. By associating the semiclassical limit with states where the corresponding particle possesses a short wavelength this apparent contradiction could be explained.

5.1

QUANTUM DYNAMICS IN PHASE SPACE

Due to the Heisenberg uncertainty relation phase space and quantum dynamics seem to be incompatible. In classical dynamics it is always theoretically possible to prepare a system knowing exactly its coordinates and momenta; for quantum systems this would be impossible. Nevertheless, by considering mean values instead of the quantum operators themselves, Ehrenfest could derive equations of motion equivalent to those known from classical Hamiltonian dynamics. Because either the coordinates are multiplicative operators and the momenta derivative operators or vice versa, the wave function can only depend on either the coordinates or the momenta, but not at the same time from canonical conjugate observables. Of course non-canonical coordinates and momenta could

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NUMERICAL QUANTUM DYNAMICS

be mixed. To map the wave function onto a phase space object, actually we should call that phase space a mock phase space4, Wigner considered instead of the wave function the density operator. The density operator or density matrix depends for systems of f degrees-of-freedom from 2f coordinates independently. Using one set of this coordinates and Fourier-transforming this set Wigner could derive a quantum object in phase space. 5.1.1

THE EHRENFEST THEOREM

The theorems relate not to the dynamical variables of quantum mechanics themselves but to their expectation values. Let  be an observable and a normalized state. The mean value of the observable at time t is

thus a complex function of t. By differentiating this equation with respect of t we obtain

with the Hamiltonian of the system under consideration. If we apply this formula to the observables and we obtain for stationary potentials, with

These two equations are called Ehrenfest’s theorem. Their form recalls that of the classical equations of motion. 5.1.2 THE WIGNER FUNCTION IN PHASE SPACE The concept of density operators was originally derived to describe mixed states. Mixed states occur when one has incomplete information about a system. For those systems one has to appeal to the concept of probability. For quantum systems this has important consequences, one of the most obvious are vanishing interference patterns. Nevertheless this is a fascinating and important problem by its own, we will restrict the following discussion to pure states, even if most of the relations presented will also hold for mixed states. The expectation or mean value of an observable  is given by

Introduction to Quantum Dynamics

therefore in a complete Hilbert space basis

21

we obtain with

and thus is called the density operator and

the density matrix. Its diagonal element

is which gives the probability to find the system in the state The von Neumann Equation. The von Neumann equation is the equation of motion for the density operator. Let us start with an arbitrary wave packet

and a complete Hilbert space basis. From the time-dependent Schrödinger equation (1.3), we obtain immediately

Multiplying from the left with Hilbert space basis, results in

and using the orthonormalization of the

respectively in

For the density matrix we obtain with

and thus for the density operator

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NUMERICAL QUANTUM DYNAMICS

This equation is called the von Neumann equation and has exactly the same structure as the classical Liouville equation. The diagonal elements of the density operator in the coordinate representation are which is just the probability of presence at position Some additional important properties of the density operator are: it is hermitian, normal and positive definite, which means for all hermitian operators

The Wigner Function in Phase Space. In the coordinate representation the density operator depends for quantum systems with f degrees-of-freedom on 2f coordinates Defining new coordinates via

we obtain the Wigner function [17]

For a better understanding let us rewrite this equation: The Fourier transform can be written as

and thus equivalently

The new coordinates and bear some similarities to relative and center of mass coordinates in many particle systems. Integrating Eq.(1.77) over the momenta gives

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23

Thus, integrating the Wigner function in phase space over the momenta results in the probability of presence in coordinate space, and vice versa, integrating over the coordinate space will lead to the probability of presence in momentum space:

Additional properties of the Wigner function in phase space can be found in [17] and many text books about quantum optics. Because the Wigner function can be obtained by Fourier transformation of the 2f-dimensional density operator, a simple computational scheme is given by discretizing the density operator in coordinate space and using a fast Fourier transformation to obtain the Wigner function [18]. Regarding the projection onto a surface-of-section allows to compare quantum Poincaré surfaces with classical surfaces. By synthesizing wave packets localized along classical structures quantum and classical dynamics can be compared and quantum effects for chaotic systems uncovered. Some examples can, e.g., found in [19] and references therein. A generalization of the Wigner function is given by the Weyl transformation. The Weyl tranformation maps an arbitrary quantum operator  onto a quantum function over phase space and is given by

Using as quantum operator the density operator will give again the Wigner function. Note, even if the quantum function above is well defined in any phase space position, this does not mean that the quantum system can be prepared for any exact phase space value. Of course the Heisenberg uncertainty principle still holds. One should not confuse the well defined behavior of quantum functions, respectively quantum operators, with preparing or measuring quantum systems. Let us consider, e.g., a wave packet. Such a wave packet has always well defined values in coordinate space and at the same time its Fourier transformed possesses exact and well defined values in momentum space. The corresponding probability of presence in coordinate respectively momentum space of this wave packet can be computed by integrating the Wigner function, see Eq.(1.79). Of course calculating the variance of the coordinate and momentum operator with this wave packet keeps the quantum uncertainty. By folding the Wigner function with a Gaussian in phase space the Husimi function [20] is defined

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NUMERICAL QUANTUM DYNAMICS

with

and a real number. Instead of using this definition it is much easier to compute the Husimi function by projection onto a coherent state [21]. In contrast to the Wigner function, the Husimi function is positive definite, which allows its interpretation as a quantum phase space probability. Once one have obtained the Wigner function for a given quantum state then one may study its evolution in time. There are two possibilities: Either one computes the evolution of the wave packet and again and again the Wigner function from the evolved wave packet or one tries to solve the equation of motion for the Wigner function directly. Computational easier is the first way. Nevertheless let us also briefly discuss the second possibility. To obtain the equation of motion for the Wigner function we have to map the von Neumann equation onto the phase space. This results in

with

the Moyal brackets [22], [23] defined by

By inserting the Taylor expansion of the sine function into this formal expression one obtains finally a series expansion of the Moyal bracket. The Moyal bracket are the quantum correspondence to the Lagrange bracket. Therefore they are in addition useful in checking, e.g., the integrability of a quantum system (see Chapt. 2). Because vanishing commutator relations for quantum operators correspond to vanishing Moyal brackets for the quantum function in phase space, this could be useful, e.g, for those quantum systems for which only the corresponding classical invariants of motion are known (see, e.g. [24]).

5.2 5.2.1

THE WKB-APPROXIMATION

THE WKB-APPROXIMATION IN COORDINATE SPACE Now we will review the WKB approximation (Wentzel-Kramers-Brillouin) to first bring out the similarity between the Schrödinger equation and the Hamilton-Jacobi equation of classical dynamics. By applying the time-independent Schrödinger equation (1.15) onto the ansatz

Introduction to Quantum Dynamics

25

we obtain

This equation is still exact. Because the last term could also be treated as a potential-like term, is called quantum potential. In the limit the equation above becomes

and

A classical conservative system holds respectively

and

Thus is interpreted as action of the corresponding classical path with Hamilton-Jacobi equation By rewriting the action of the classical system in terms of the potential we obtain

and hence one condition for the validity of the WKB-approximation can be translated into a criteria of the potential: the above approximation will be satisfied for slowly varying potentials. This will not be hold at turning points, which will become more obvious after discussing Eq.(1.85). From we obtain and thus Eq.(1.85) becomes

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NUMERICAL QUANTUM DYNAMICS

and this equation can be immediately solved:

At the turning points of the classical trajectory in coordinate space the canonical conjugate momentum vanishes and thus the wave amplitude function A(q) diverges. 5.2.2 THE WKB-APPROXIMATION IN MOMENTUM SPACE Classically coordinates and canonical conjugate momenta can interchange their rôle by a canonical transformation. Similar, quantum coordinate space is connected with the momentum space by a Fourier transformation. Therefore Maslov solved the singularity problem mentioned above by considering the WKB-ansatz in momentum space. The Hamiltonian in momentum space is given by

and the WKB-ansatz by

In momentum space the kinetic energy operator becomes multiplicative and the potential operator a derivative operator. Therefore but this would not hold for the potential operator. Let us assume Thus we obtain from the potential operator up to orders in

and with

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27

only two contributions remain due to the restrictions in the summation above. In we get and thus and hence and In we have which results in and therefore Putting these pieces all together we obtain

and thus

with

Therefore we get in analogy to the WKB-equations in coordinate space the following WKB-equations in momentum space:

and

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With

and

we obtain

and therefore Thus again we get divergencies, but now at the turning points in momentum space. Hence we have two solutions, both diverging, but fortunately at different places in phase space. Therefore the essential idea is to combine the solution in coordinate space with the Fourier transformed solution of the momentum space, which will lead to a coordinate space solution with divergences at different points in phase space. Note, because both solutions are only approximations they are not Fourier transformed solutions to each other. Let be the WKB-soIution in coordinate space at point q(t) and the WKB-solution in momentum space. Its Fourier transformation is given by

Because we are only interested in lowest order in it is sufficient to compute this integral in stationary phase approximation. The solutions of the WKB approximation are given by periodic trajectories in the corresponding classical system. A periodic trajectory can run through the same point in coordinate space several times. Therefore we get our solution in coordinate space by an overlap of all contributions for fixed q, but at different times. Of course this holds also equivalently in momentum space. If we would only sum over all those contributions we would still have to fight with the original singularity. Therefore we multiply our original solution in coordinate space with a suitable selected function and our Fourier transformed solution with a second function The main task of this function is that it becomes zero at the singularities, additionally it is a periodic function with the period of the classical trajectory under consideration and the sum of both functions equals one:

where denotes the singularities of the coordinate and the singularities of the momentum space. Thus we obtain two new solutions

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29

and

and our final singularity-free solution becomes

5.2.3 THE CONNECTION FORMULAS As described above both in the momentum and in the coordinate formulation of the WKB-method singularities occur at the classical turning points. This problem could be solved by combining the solution in coordinate space with the Fourier transformed momentum space solution, or, if we should be interested in a solution in momentum space, vice versa. Obtaining the solution in momentum space and Fourier transforming it might be often an unjustified laborious solution in spite of its approximate character. Nevertheless the solution presented above holds in any dimension and is rather general. A simpler way for one-dimensional and radial systems is given by the connection formula, which we will briefly discuss for bound states and one-dimensional systems. Let us assume, that around the turning point the potential variation is small such that the potential can be linearly approximated, with the tangent at the singularity and E the energy, because at the turning point in coordinate space the kinetic energy vanishes and thus the potential energy equals the total energy. Thus in the vicinity of the turning point the Schrödinger equation becomes

and with the variable transformation

we obtain

The Airy functions are the solutions of this equation. Because we are interested in bound states we are only interested in those Airy functions, which goes to zero for Example. In the following we will discuss the formulas derived above by a simple example. Fig.(1.2) shows a potential well with turning points To derive our solution we devide the space into the turning points and three regions. Outside the well: and and the area inside the well For

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NUMERICAL QUANTUM DYNAMICS

the solution outside the well the momentum becomes imaginary because the potential becomes larger than the energy. Thus we get from our WKB-ansatz (1.84) outside the interior region

For both and the wave function has to vanish. Hence we obtain on the lefthand side the ’positive’ and on the righthand side the ’negative’ sign in the exponential function. These solutions have to connect to the oscillatory solution inside the well. The ’inside’ solution is given by a linear combination of the two solutions of the action S(q) in Eq.(1.86). This inside solution will be given by the Airy function which goes smoothly through the turning points. In the neighborhood of the turning point we get from the outside

and therefore

and inside the well

The asymptotic solution of the Airy function is

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31

and thus the WKB solution inside the well will only match this Airy solution if we take a linear combination of exp and exp such that

Next, let us discuss the turning point analysis we get

on the lefthand side. By a similar

These two equations represent the same area in space and thus they have to be identical up to a normalization constant N. Therefore we obtain

and thus

By the analysis above we have derived a quantum condition

for a bound state in a potential well with smoothly varying turning points, Because outside the well the wave function converges exponentially to zero the quantum number gives the number of nodes of the wave function, see Eq.(1.97). The WKB wave function is given by

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Further aspects of the WKB solution, especially in the context of EBK6 and torus quantization, can be found in the very readable book of Brack and Bhaduri [15].

Notes l Those readers able to understand German are recommended to read Schrödinger’s original papers – masterpieces in science and language! (See, e.g., E. Schrödinger, Von der Mikromechanik zur Makromechanik, Naturwissenschaften (1926), 14, 664). 2 Schrödinger published his results 1926 a few months after Pauli derived the hydrogen spectrum based on Heisenberg’s “matrix mechanics”. 3 The names bra and ket are built from bra-c-ket:

4 In phase space each position has a well defined value. But for the quantum system this does not allow us to prepare a single point in phase space or to detect a single point in phase space – of course the uncertainty principle still holds. Therefore we call the phase space in the quantum picture a “mock phase space”. stands for trace. 5 6 EBK = Einstein-Brillouin-Keller

References [1] Cohen-Tannoudji, C., Diu, B. and Laloë, F. (1977). Quantum Mechanics I, II. John Wiley & Sons, New York [2] Dirac, P. A. M. (1967). The Principle of Quantum Mechanics (4th edition). Oxford Science Publ., Claredon Press, Oxford [3] Merzbacher, E. (1970). Quantum mechanics. John Wiley & Sons, New York [4] Planck, M. (1995). Die Ableitung der Strahlungsgesetze. (A collection of reprints of M. Planck published from 1895 – 1901, in German.) Verlag Harri Deutsch, Frankfurt am Main [5] Simon, B. (1976). “The bound states of weakly coupled Schrödinger operators in one and two dimensions,” Ann. Phys. (N.Y.) 97, 279–288

References

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[6] Calogero, F. (1965). “Sufficient conditions for an attractive potential to possess bound states,” J. Math. Phys. 6, 161–164 [7] Chadan, K., Kobayashi, R. and Stubbe, J. (1996). “Generalization of the Calogero-Cohn bound on the number of bound states,” J. Math. Phys. 37, 1106–1114 [8] Ho, Y. K. (1983). “The method of complex coordinate rotation and its applications to atomic collision processes,” Phys. Rep. 99, 1 – 68 [9] Moiseyev, N. (1998). “Quantum theory of resonances: Calculating energies, widths and cross-sections by complex scaling,” Phys. Rep. 302, 211– 296 [10] Biedenharn, L. C. and Louck, J. D. (1981). Angular momentum in quantum physics. Addison-Wesley, Reading [11] Regge, T. (1958). “Symmetry properties of Clebsch-Gordan’s coefficients”, Nuovo Cim. 10, 544–546 [12] Edmonds, A. R. (1957). Angular momentum in quantum dynamics Princeton University Press, New York [13] Goldberg, A., Schey, H. M. and Schwartz J. L. (1967). “Computergenerated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. Journ. of Phys. 35, 177 – 186 [14] Faßbinder, P., Schweizer, W. and Uzer, T (1997). “Numerical simulation of electronic wavepacket evolution,” Phys. Rev. A 56, 3626 – 3629 [15] Brack, M. and Bhaduri, R. (1997). Semiclassical Physics AddisonWesley, Reading [16] Gao, B. (1999). “Breakdown of Bohr’s Correspondence Principle,” Phys. Rev. Lett. 83, 4225 – 4228 [17] Hillery, M., O’Connell, R. F., Scully, M. O., and Wigner, E. P. (1984). “Distributions Functions in Physics: Fundamentals”, Phys. Reports 106, 121 – 167 [18] Schweizer, W., Schaich, M., Jans, W., and Ruder, H. (1994). “Classical chaos and quantal Wigner distributions for the diamagnetic H-atom”, Phys. Lett. A 189, 64 – 71 [19] Schweizer, W., Jans, W., and User, T (1998). “Optimal Localization of Wave Packets on Invariant Structures”, Phys. Rev. A58, 1382 – 1388; (1998) “Wave Packet Evolution along Periodic Structures of Classical Dynamics” ibid. A60, 1414 – 1419

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[20] Takahashi, K. (1986). “Wigner and Husimi function in quantum mechanics”, J. Phys. Soc. Jap. 55, 762 – 779 [21] Groh, G., Korsch, H. J., and Schweizer, W. (1998). “Phase space entropies and global quantum phase space organisation: A two-dimensional anharmonic system”, J. Phys. A 31, 6897 – 6910 [22] Moyal, E. J. (1949). “Quantum Mechanics as a Statistical Theory”, Proc. Cambridge Phil. Soc. 45, 99 – 124 [23] Takabayasi, T. (1954). “The Formulation of Quantum Mechanics in Terms of Ensemble in Phase Space” Prog. Theor. Phys. 11, 341 – 373 [24] Hietarinta, J. (1984). “Classical versus quantum integrability”, J. Math. Phys. 25, 1833 – 1840

Chapter 2 SEPARABILITY

The Schrödinger equation is a partial differential equation and in the coordinate representation its kinetic energy operator a Laplacian. Thus separability is a great simplification of the numerical task. Partial differential equations involving the three-dimensional Laplacian operator are known to be separable in eleven different coordinate systems [1, 2]. We will label all eleven coordinate systems in section two, but discuss in detail only some of them. In section three we will briefly discuss coordinate singularities exampled by the Coulomb- and the centrifugal singularity. Each separable system is integrable but not vice versa. Conservative systems can be divided into two types, integrable and non-integrable. Integrable systems have as many independent integrals of motion as they have degrees-of-freedom. Thus there is a qualitative difference between integrable and non-integrable systems. A decisive stimulus came from the discovery that even simple systems exhibit a non-integrable - or more popular - chaotic behavior. Since this is an exciting phenomenon, this is where we will begin our discussion.

1.

CLASSICAL AND QUANTUM INTEGRABILITY

Classical integrability. Let us consider a classical system with f degreesof-freedom. Its phase space has 2f dimensions. A phase space coordinate is denoted by with the position vector and its canonical conjugate momentum vector A constant of motion is a function over phase space whose Poisson bracket with the Hamiltonian vanishes

35

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A conservative Hamiltonian system with f degrees-of-freedom is called integrable if and only if there exist f analytical functions with 1. the analytical functions are analytically independent, which means are linear independent vectors in phase space 2. the analytical functions 3. all

are constants of motion

are in involution, which means the mutual Poisson brackets vanish,

If a system is integrable all orbits lie on f-dimensional surfaces in the 2fdimensional phase space and there are no internal resonances leading to chaos. Due to Noether’s theorem constants of motion result from symmetries. For more details see, e.g., [3,4], Quantum integrability. Classical integrability is our model for quantum integrability and thus integrability for quantum systems is defined in an analogous manner [5, 6]. A conservative quantum system with f degrees-of-freedom is called integrable if and only if there exist f globally defined hermitian operators whose mutual commutators vanish

Independent constants of motion imprison classical trajectories in lower dimensional subspaces of the 2f-dimensional phase space. Hence, for each constant of motion which becomes obsolete the dimension of the dynamically accessible space will be increased by one. Due to the Heisenberg uncertainty principle trajectories do not exist in quantum dynamics. Loosely speaking quantum numbers will take their rôle. Observables in quantum dynamics are given by hermitian operators. Therefore integrability of a f-dimensional Hamiltonian system requires the existence of f commuting observables Mutually commuting observables have common eigenvectors and their eigenvalues can serve as quantum-labels for the wave functions. Therefore the number of constants of motion equals the number of commuting observables and corresponds to the number of conserved or good quantum numbers . For separable systems the eigenvalues (separation constants) of each of the f unidimensional differential equations can be used to label the eigenfunctions and hence serve as quantum numbers. If the number of commuting observables decreases, the number of good quantum numbers will also decrease and the quantum system becomes non-integrable. A simple example is given by spherical symmetric systems. In this case the Hamiltonian will commute with the angular momentum operators

Separability

37

1, 2, 3 and because of the mutual commuting observables are and one angular momentum component, e.g., Ergo, eigenvectors of the system can be labeled by the corresponding eigenvalues: the energy E, from which we get the principal quantum number the angular momentum l obtained from and the magnetic quantum number this holds, e.g., for the hydrogen atom. If the hydrogen atom is placed in an external magnetic field the spherical symmetry will be perturbed and l is no longer a conserved quantum number. Because l is no longer conserved degeneracy of states with equal parity and equal magnetic quantum number is now forbidden, and hence avoided crossings or level repulsion occur. An example is shown in Fig.(2.1) (see also Chapt.6.?.). Of course this holds in general for any quantum system. Thus if we compare, e.g., the statistical properties of spectra for quantum integrable and quantum non-integrable systems there will be a qualitative difference. Those degeneracies which were related to quantum numbers existing only in the integrable limit, will become forbidden for the non-integrable quantum system. One of the most established statistical investigations in quantumchaology is the nearest neighbor statistic. The basic idea is to compute the probability to find a definite spacing between any two neighboring energy levels. Because in general the density of states is energy dependent, first the quantum spectrum under consideration have to be transformed to a spectrum with a constant

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NUMERICAL QUANTUM DYNAMICS

averaged density of states, the so-called defolded spectrum. Then in this defolded spectrum the probability to find a certain spacing, S, between any two neighboring eigenlevels in the interval will be computed. For integrable systems the highest probability is to find degeneracy and the histogram is best fitted by a Poisson distribution

For quantum non-integrable systems degeneracy is forbidden and hence for vanishing spacing the nearest neighbor distribution function, will vanish. The correct distribution will depend on discrete symmetry properties. For time-invariant non-integrable quantum systems this will be a Wigner distribution

An example is presented in Fig.(2.2). For more details about quantum chaos and statistics see, e.g., [8].

2.

SEPARABILITY IN THREE DIMENSIONS

Integrability of a 3-dimensional Hamiltonian system requires the existence of 3 commuting observables. For a three dimensional system it can be shown that the Laplace-Beltrami operator can be separated in exactly 11 different curvilinear coordinates. For each of these coordinates the potential has to fulfill certain properties that the Schrödinger equation becomes separable.

Separability

For all orthogonal curvilinear coordinates ing conditions hold:

39

in three dimensions the follow-

with and is called Laplace-Beltrami operator, and these equations are the most relevant expressions in curvilinear coordinates for quantum systems: The components of the momentum operator are given by the nabla-operator, the kinetic energy by the Laplace-Beltrami operator and to obtain, e.g., the probability of presence the volume element in the new coordinates is needed. The only remaining task is to select the coordinate system for which, if any, the Schrödinger equation becomes separable. This is the case if the potential can be written as

Using Eq.(2.7) it can be proofed easily if and for which coordinates a given one-particle quantum system is separable. (1) Cartesian coordinates.

A quantum system becomes separable in Cartesian coordinates if the potential can be written as With a product ansatz for the wave function

we obtain immediately

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NUMERICAL QUANTUM DYNAMICS

with E the energy, and and separation constants. Separable in Cartesian coordinates are, e.g., the 3d-harmonic oscillator. The scheme of the derivation of the one-dimensional differential equations above for quantum systems separable in Cartesian coordinates, holds also for curvilinear coordinates. The first step is to derive the Laplace-Beltrami operator in the new coordinates and then to act with the Schrödinger equation in the new coordinates on a product ansatz to derive the uni-dimensional differential equations. In the following list we will discuss only for some examples how the Laplace-Beltrami operator looks like and for which type of potentials the Schrödinger equation becomes separable, but leave the complete derivation to the reader. (2) Spherical coordinates.

In spherical coordinates the metric tensor

is

and the Laplace-Beltrami operator

A quantum system becomes separable in spherical coordinates, if the potential fulfills

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41

Of course this holds for all systems with spherical symmetry, e.g., the spherical oscillator, the Kepler system, the phenomenological Alkali metal potentials, the Morse oscillator and the Yukawa potential to name only a few. (Many examples are discussed in [10].) These quantum systems possess a central potential. Since the angular momentum operator in spherical coordinates is given by Eq.(1.37c) and the Laplace-Beltrami operator by Eq.(2.11) the central field Hamiltonian can be written as

with the particle mass or (for two particle systems) reduced mass. Labeling the eigenstates by the angular momentum quantum number l, the magnetic quantum number and an additional quantum number related to the energy, we obtain the Hamiltonian eigenvalue equation

With a product ansatz for the wave function

the radial Schrödinger equation reads

This solution depends on the angular momentum l and thus the radial wave function has to be labeled by both, the quantum number n and the total angular momentum l. For bound states it is convenient to rewrite the radial Schrbödinger equation. Using instead of

the radial contribution to the Laplace-Beltrami operator becomes

and thus the radial Schrödinger equation is actually simpler in terms of

and becomes similar the uni-dimensional Schrödinger equation.

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NUMERICAL QUANTUM DYNAMICS

(3) Cylindrical coordinates.

In cylindrical coordinates the metric tensor

is

and the Laplace-Beltrami operator

A quantum system becomes separable in cylindrical coordinates, if the potential fulfills

An important example for a system which possesses cylindrical symmetry is the free electron in a homogeneous magnetic field. (4) Parabolic coordinates.

The metric tensor

becomes in parabolic coordinates

and the Laplace-Beltrami operator

A quantum system becomes separable in parabolic coordinates, if

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43

Two important applications for parabolic coordinates are Rutherford scattering and the Stark effect (hydrogen atom in an external homogeneous electric field), which we will discuss briefly. The Hamiltonian of the hydrogen atom in an external electric field F in atomic units reads

Hence in parabolic coordinates we obtain

and with a product ansatz

with and the separation constants. Because is connected with the charge it is also called fractional charge. For hydrogen-like ions with charge Z the Coulomb potential becomes and thus the two separation constants and holds respectively in Eq.(2.27c) we get instead of Due to the electric field tunelling becomes possible and thus all former bound states become resonances. Elliptic differential equations allow only bound states. Therefore the mathematical structure of the Kepler equation in an additional electric field had to change: Eq.(2.27b) is of elliptic type hence supports bound states, whereas Eq.(2.27c) is of hyperbolic type, ergo possesses only continuum states. For more details about the hydrogen Stark effect see [11].

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(5) Elliptic cylindrical coordinates.

(6) Prolate spheroidal coordinates.

(7) Oblate spheroidal coordinates.

(8) Semiparabolic cylindrical coordinates.

(9) Paraboloidal coordinates.

(10) Ellipsoidal coordinates.

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45

(11) Cone coordinates.

Semiparabolic coordinates. The semiparabolic coordinates can be derived from the semiparabolic cylindrical coordinates and the spherical coordinates and therefore are not independent. Nevertheless they play an important role in lifting the Coulomb singularity, see section 3.1, and therefore we will discuss this coordinate system

in addition. The metric tensor of the semiparabolic coordinates is

and the Laplace-Beltrami operator becomes

Quantum systems are separable in semiparabolic coordinates if the potential reads

Fur further discussions see sect.(3.1).

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NUMERICAL QUANTUM DYNAMICS

Elliptic coordinates.

The only molecule for which the Schrödinger equation can be solved exactly is the hydrogen molecule-ion, in Born-Oppenheimer approximation. The basic idea of the Born-Oppenheimer approximation is that we regard the nuclei frozen in a single arrangement and solve the Schrödinger equation for the electron moving in this stationary Coulomb potential they generate. This approximation is justified by the different masses of the electron and the nuclei. The mass-relation between proton and electron is approximately 1 : 1836. The geometry of such a molecule-ion is presented in Fig.(2.3). Transforming the Schrödinger equation to elliptic coordinates results in a separable differential equation.

The Hamiltonian of the electron motion in Born-Oppenheimer approximation is

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47

with

and the energy is a parametric function of the nuclei distance R. Thus the correct energy is given by the minimum of By introducing an angle and coordinates

we arrive at

and these are exactly the elliptic coordinates defined above. The Laplace-Beltrami operator becomes

and with the ansatz and some arithmetic we obtain finally

and

with separation constants. Solving these equations, we find a discrete spectrum for each value of R. We will further discuss this example in Sect.3 in context of variational calculations.

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NUMERICAL QUANTUM DYNAMICS

Polar coordinates.

Polar coordinates are often used to partially separate higher dimensional partial differential equations. Due to their similarity to spherical, cylindrical and hyperspherical coordinates we will not discuss them in detail here (see Chapt, 3.2). Jacobi coordinates. Jacobi and hyperspherical coordinates go beyond the above-discussed 3-dimensional coordinates. Jacobi coordinates are used for N-particle systems. The following discussion will be restricted to systems with particle of identical mass. Otherwise mass-weighted coordinates have to be used. Jacobi coordinates are used typically for n-electron systems in atomic physics or n-nucleon systems in nuclear physics. Hence the particle are fermions and symmetry properties like the Pauli principle are simplified. For two-particle systems the Jacobi coordinates are identical with relative and center-of-mass coordinates

with the individual particle coordinates. Jacobi coordinates for an Nparticle system are defined as the relative distance between the th particle and the center of mass of the particle

The kinetic energy

with

in Jacobi coordinates reads

the particle mass.

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49

Hyperspherical coordinates. Hyperspherical coordinates generalize polar and spherical coordinates to N dimensions via

thus hyperspherical coordinates hold

3.

COORDINATES AND SINGULARITIES

The successful story of quantum dynamics began by explaining the hydrogen spectrum back in the nineteen-twenties. This is the Kepler problem in quantum mechanics, the motion of an electron in a There is a slight problem in the radial Schrödinger equation, the singularity at the origin, A standard textbook example is to remove this singularity by an ansatz for the wave function with for the regular and for the irregular solution. In this section we will discuss coordinate systems for which the Schrödinger equation of the Kepler system will be mapped on a singularity-free form.

3.1

THE KEPLER SYSTEM IN SEMIPARABOLIC COORDINATES

It is of interest that different information of a system are uncovered by using different coordinate systems. The hydrogen atom is usually treated in spherical coordinates, because in this coordinates wave functions and energies can be conveniently computed. By using semiparabolic coordinates the hydrogen atom becomes similar to an harmonic oscillator. Let us start with the KeplerHamiltonian

with the radial distance of the electron to the origin and E the eigenenergy. We obtain in semiparabolic coordinates with and Eq.(2.36)

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NUMERICAL QUANTUM DYNAMICS

and with ansatz

Therefore the Kepler system is separable in semiparabolic coordinates. For bound states the energy becomes negative and a real number and for continuous solutions is imaginary. Let us first consider the subspace For we obtain two similar ordinary differential equations of the kind

and the separation constants. This is the differential equation for a two-dimensional harmonic oscillator in polar coordinates. With

we obtain for

and by the variable transformation

the Kummer equation [12]. The solution of the Kummer equation1 are the hypergeometric confluent series, which will converge for Thus we obtain and for the hydrogen energy for

Therefore we have of course recovered the equation for the hydrogen energy. Because are positive integers, the energy ground state is again –1/2 and the degeneracy for states again with Thus the description above is equivalent to the Kepler description for but the Coulomb singularity is removed.

Separability

3.2

51

THE KUSTAANHEIMO-STIEFEL COORDINATES

Circular oscillator.

The Schrödinger equation for the circular oscillator is

In polar coordinates

we obtain for the metric tensor Beltrami operator

and thus for the Laplace -

Together with the separation ansatz

this leads to

for the Schrödinger equation of the circular oscillator in polar coordinates. (We have this equation already discussed for But our focus is not on the two dimensional oscillator. More details about the two-dimensional harmonic oscillator can be found in most text books on quantum mechanics.)

The Kustaanheimo-Stiefel transformation. Eq.(2.59) and Eq.(2.52) are equivalent in structure, but Eq.(2.52) depends on two and not only on one coordinate. Therefore we will briefly discuss the four dimensional harmonic oscillator. The Hamiltonian of the degenerate four dimensional oscillator is

By the following bi-circular transformation

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NUMERICAL QUANTUM DYNAMICS

we obtain for the Hamiltonian of the four dimensional harmonic oscillator with the ansatz,

Compared to Eq.(2.52) there are two shortcomings: The harmonic oscillator energy E could be in general unequal 2, and more important the two angular momentum quantum number and are not necessarily equal. The first problem could be solved easily, by restricting to those values for which E equals 2, The second problem could be solved by introducing an additional constraint. Restricting the solution above to leads to the scleronom constraint

Transformation (2.60) together with the transformation from the semiparabolic coordinates to the spherical coordinates under the constraint Eq.(2.63) is called Kustaanheimo-Stiefel transformation and the coordinates of Eq.(2.60) Kustaanheimo-Stiefel coordinates [13] [14], By this transformation, both the Coulomb singularity and the centrifugal singularity are lifted. Of course there is no necessity to introduce the intermediate semiparabolic transformation and one could also transform directly the spherical coordinates to the four dimensional space together with the constraint, but the intermediate semiparabolic transformation uncovers more clearly the physical structure behind.

Notes 1 For the two-dimensional harmonic oscillator we would obtain

References [1] Morse, P. M. and Feshbach, H. (1953). Methods of Theoretical Physics Mc-Graw-Hill, NewYork [2] Miller, W. (1968). Lie theory and special functions Academic Press, New York

References

53

[3] Lichtenberg, A. J. and Lieberman, M. A. (1983). Regular and stochastic motion Springer-Verlag Heidelberg [4] Arnold, V. I, (1979). Mathematical methods of classical physics SpringerVerlag Heidelberg [5] Hietarinta, J. (1982), “Quantum integrability is not a trivial consequence of classical integrability”, Phys. Lett. A93, 55 – 57 [6] Hietarinta, J. (1984). “Classical versus quantum integrability”, J. Math. Phys. 25, 1833–1840 [7] Schweizer, W. (1995). Das diamagnetische Wasserstoffatom. Ein Beispiel für Chaos in der Quantenmechanik, Verlag Hard Deutsch Frankfurt am Main [8] Haake, F. (1991). Quantum signatures of chaos Springer-Verlag Berlin [9] Jans, W., Monteiro, T., Schweizer, W. and Dando, P. (1993). “Phase-space distributions and spectral properties for non-hydroenic atoms in magnetic fields” J. Phys. A26, 3187 – 3200 [10] Flügge, S. (1994). Practical Quantum mechanics (2nd printing) SpringerVerlag Berlin [11] Harmin, D. A. (1982). “Theory of the Stark effect” Phys. Rev. A 26, 2656 – 2681 [12] Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions. Dover Publications, Inc., New York [13] Kustaanheimo, P. and Stiefel, E. (1964). “Pertubation theory of Kepler motion based on spinor regularization”, Journal für Mathematik, 204 – 219 [14] Cahill, E. (1990). “The Kustaanheimo-Stiefel transformation applied to the hydrogen atom: using the constraint equation and resolving a wavefunction discrepancy”, J. Phys. A 23, 1519–1522

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Chapter 3 APPROXIMATION BY PERTURBATION

The necessary step in dealing with real-world quantum systems is to develop approximation techniques. The next two chapters are devoted to perturbation and variational techniques. The term ‘perturbation’ and the idea of this method were introduced in quantum dynamics by analogy with the perturbation method of classical mechanics. These perturbations can be imposed externally or they may represent interactions within the system itself. This numerical technique works so long as we can treat the perturbation as small - which immediately rises the question, what is small? There are two kinds of perturbation, time-dependent and time-independent. First we will discuss time-independent methods and illustrate its application by some examples. We will start with the Rayleigh-Schrödinger perturbation theory. With the advent of computers, this seems to be old-fashioned. But perturbation theory retains its usefulness because of a certain physical clarity it gives us, and not only as testing method of more advanced numerical techniques in certain parameter limits. Therefore whenever it is applicable, low-order perturbation should be used to gain additional physical insight, and more advanced numerical techniques to get accurate numbers. In the first section we will develop the Rayleigh-Schrödinger perturbation technique which can be also found in many quantum mechanic textbooks, see, e.g., [1], [2]. We will discuss both, ordinary and degenerate perturbation theory. The next two sections will be devoted to more advanced perturbation techniques, the 1/N-shift theory and the approximative symmetry. Perturbation acting on a system are very non-stationary. Thus we will close this chapter with a brief discussion of time-dependent perturbation theory. 55

56

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1.

THE RAYLEIGH-SCHRÖDINGER PERTURBATION THEORY NONDEGENERATE STATES

1.1

Let us first consider the simplest case in which the Hamiltonian of the quantum system does not depend explicitly on the time. Suppose we know the eigenvalues and the eigenstates of what seems from physical considerations the mayor part of the Hamiltonian, called

where we will assume additionally in this subsection, that is nondegenerate. We further assume that the total Hamiltonian can be written as

where we add the formal parameter for convenience. For the Hamiltonian becomes the unperturbed Hamiltonian, and for the Hamiltonian goes over to the Hamiltonian proper for the system under consideration. Let us denote the eigenstates of the complete Hamiltonian as with eigenvalues We seek a solution of this equation by a power series expansion of and We will assume, that the unperturbed eigenstates form a complete orthonormal set. Thus we start with the ansatz

Then we substitute this ansatz into the Schrödinger equation (3.3), which gives

In order that equation 3.5 can be satisfied for arbitrary the coefficient of each power of on both sides must be equal. Thus we obtain for

Approximation by Perturbation Techniques

57

Multiplying with the bra from left and using the orthogonality, we obtain the first order correction to the energy

This is a very important formula. It states that the first order energy correction to any unperturbed state is just the expectation value of the perturbing Hamiltonian in the unperturbed state under consideration. But this is not the only information hidden in the equation above. By multiplying with the bra from the left we get

Therefore the expansion coefficients in first order perturbation theory depends on both, the the non-vanishing off-diagonal matrix elements, and on the energy difference between the states under consideration of the unperturbed system. Therefore if all off-diagonal matrix elements are of the same order neighboring states mix more than far away ones. In addition, we can now see that "small" for a perturbation depends from the magnitude of the matrix elements and from the level separation of the unperturbed system due to the denominator of Eq.(3.8). In 2nd order we obtain

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and thus by multiplying from the left with the bra bra

respectively with the

and

and finally

Therefore if is the ground state, that is the state of lowest energy, then the denominator in Eq.(3.9) is always negative. Due to the hermiticity of the Hamiltonian, the nominator of Eq.(3.9) is always positive. Hence the ground state energy correction in second order perturbation theory is always negative. Energy correction - summary.

With

we get for the first four orders in the energy correction

Approximation by Perturbation Techniques

59

1.1.1 ACCELERATION TECHNIQUES By low order perturbation theory we could gain additional physical insight, as already mentioned. Of course acceleration techniques are only useful for high order perturbation expansions. In the next chapter we will briefly mention some of these methods. In many situations discretization and finite element techniques or quantum Monte Carlo methods are more accurate and due to the power of modern low-priced computers even easier available. A discussion of large order perturbation theory, especially discussing semiconvergent and divergent series for quantum systems can be found, e.g., in [3] and a brief numerical example for series in [4]. A quadratically convergent series converges in general faster than a linear convergent series. Nevertheless there are several tricks for accelerating the rate of convergence of a series. Given a (convergent) series with

the partial sums. Thus lim

If there exist a transformation

will converge quicker than and if lim the rate of convergence of will be higher than the rate of convergence of to the limit A. Aitken’s process is one of the standard methods to accelerate the convergence of series. For the following construction let us assume for simplification that have the same sign and that is sufficiently large, that

Thus

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and hence which gives immediately

Based on this simple derivation we define Aitken’s up the convergence of series by

techniques.

transformation to speed

is a simple example, which shows the power of accelerating and given by Eq.(3.15). Then

but Of course one should never forget, that computing necessitates to know and therefore one should always compare and not partial sums of the same order. But nevertheless converges significantly quicker than to 1.

1.2

DEGENERATE STATES

We now assume that the eigenvalue whose perturbation we want to study is degenerate. We denote the orthonormal eigenstates by with Without degeneracy we started with an ansatz

Because the n-th level is degenerate with a multiplicity the former nondegenerate states in the expansion above have to be replaced by a unknown linear combination of the eigenstates

Without degeneracy we got from

Approximation by Perturbation Techniques

61

but for degenerate states this will lead to a homogenous linear system of equations for the expansion coefficients

Hence the eigenenergies in first order perturbation theory are obtained by the nodes of the characteristic polynomial, respectively by the zeros of the determinant In first order degenerate perturbation theory we obtain a linear system of equations whose roots with eigenvectors will give us the eigenstate in zero order by

and the eigenenergy in first order by

Example. Let us examine more closely the case of a two-fold degenerate system. For simplicity we will suppress the level quantum number Thus as ansatz for the zero order eigenstates we choose

and thus orthonormalization will be conserved. Substituting this ansatz in Eq.(3.17) we obtain the system of homogeneous equations

with determinant

The eigenvalues are given by the zeros of the

hence in first order perturbation theory we get for the energy shift

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and the eigenfunctions are given via

Therefore the former degenerate eigenstates become non-degenerate under a perturbation with energy distance even for an arbitrary small perturbation. Thus the level structure is qualitatively changed under a small perturbation. Without perturbation the levels are degenerate and a quantum number for labeling these degenerate states exist. A simple example is given by the hydrogen atom, for which the levels are degenerate with respect to the angular momentum and the magnetic quantum number Both quantum numbers are associated with operators commuting with the Hamiltonian of the system. Under some perturbations these operators are no longer conserved operators and thus the corresponding quantum numbers are no longer "good" quantum numbers. Therefore there are no additional quantum labels for distinguishing degenerate levels and degeneracy becomes forbidden. Even under an arbitrary small perturbation degeneracy becomes forbidden. This effect is called exponentially avoided crossing or level repulsion. An example is given by the hydrogen atom in external magnetic and electric fields. Under a magnetic field pointing into the z-direction, the z-component of the angular momentum and the z-parity is conserved. Thus if we plot the Hamiltonian and the z-component of the spin) degeneracy occurs for eigenstates with the same m-quantum number but different z-parity . An example is shown in Figs.(3.1) and (3.2). Note, the quantum number is no longer conserved due to the diamagnetic contribution. For the hydrogen atom in parallel electric and magnetic fields the symmetry with respect to reflection on the is broken by the electric field and hence the is no longer conserved2, thus degeneracy with respect to becomes forbidden and an avoided crossing occurs. In Fig.(3.1) the electric field strength is vanishing, the parity conserved and hence crossings of eigenstates with different parity are allowed. In Fig.(3.2) some results for a non-vanishing electric field strength are shown. The degeneracy is broken by a small avoided crossing. One important aspect is the behavior of the eigenstates. In case of an allowed crossing, Fig.(3.1), the wave functions run without any distortion through this crossing. In Fig.(3.2) the states are interacting with each other. Far away from the crossing the wave functions are similar to the ones without an additional electric field. Close to the energy value at which the avoided crossing occurs, both wave functions are distorted and mixed with each other.

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63

By the simple hydrogen atom in external fields we have a very graphic example, showing the interaction of neighboring eigenstates for non-integrable quantum systems, an effect, which holds in general and which results also in

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a completely different statistical behavior of the energy eigenvalues (see, e.g., [6]). For integrable quantum systems (see chapter 2 and Fig.(2.1)) degeneracy is allowed and hence the most probable energy distance between neighboring en-

Approximation by Perturbation Techniques

65

ergy levels is zero. Thus the probability of finding a certain energy distance for a spectrum with constant density follows a Poisson distribution. The question arises, how does this behavior change if degeneracy becomes forbidden due to an additional perturbation. For simplification the following discussion will be restricted to a two-level system invariant under time-reflection and rotation. To derive the probability of finding a certain energy distance between neighboring energy levels let us first ask ‘what is the probability for finding a certain Hamiltonian matrix’ ? The Hamiltonian matrix of our two level system is given by

with real and The matrix is given by their elements, thus the probability of finding a certain matrix H, P(H), is given by the probability for finding certain matrix elements, and thus

This result has to be invariant under unitary basis transformation and hence under an infinitesimal basis transformation:

The phrase "infinitesimal" means that terms in

are negligible, hence

and with

and thus

Because of the infinitesimal transformation the probability of finding the new primed matrix elements are given in first order Taylor expansion, resulting in

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Both matrices are equivalent and therefore the probability of finding the new unitary transformed matrix equals the probability of finding the original matrix. (More technical speaking, we are only interested in the ensemble probability, which is independent under the selected basis.)

Thus

which results in

The lefthand side depends only on contributions from the "12" elements and the righthand side is completely independent from those "12" elements. Therefore this equation have to be constant and we obtain

The probability of finding any matrix element is one and thus

Repeating the same game for the "11" and "22" elements gives

Approximation by Perturbation Techniques

67

and finally

with A = 0. From the beginning we were only interested in the probability of finding a certain energy distance between neighboring levels, but not in the matrix elements it selves. Therefore let us now compute the probability function as a function of the energies and the mixing angle given by Eq.(3.22) and Eq.(3.23):

The matrix elements are given by

and the Jacobi determinant by

Therefore we obtain for the probability of finding a matrix H

and as a function of the physical parameters energy and mixing angle

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NUMERICAL QUANTUM DYNAMICS

Hence the probability of finding a certain energy spacing by

is given

With the averaged energy spacing

we arrive finally at

and this probability function is called Wigner function. The probability for finding a degenerate level, vanishes. Usually the averaged energy spacing is transformed to D = 1 and the energy spacing is denoted by "S". Because this probability distribution is by construction invariant under orthogonal transformation it is also called a Gaussian orthogonal ensemble, abbreviated by GOE. Other important probability functions are

where the names reflect the invariance of the Hamiltonian matrix under certain transformations. Of course this transformation properties are due to certain physical qualities like time invariance and so forth. For more details see, e.g., [7]. Remember, that for integrable quantum systems degeneracy is allowed and the energy spacing follows a Possion distribution P(S) = e x p ( – S ) . 1.2.1 QUADRATIC AND LINEAR STARK EFFECT To illustrate the rôle of degeneracy in perturbation theory, we will consider the effect of an external electric field on the energy levels of the hydrogen atom, the Stark effect.3 The Hamiltonian for a hydrogen atom in an electric field with field axis pointing into the reads (in atomic units)

The hydrogen ground state is nondegenerate and thus the energy correction in first order perturbation theory is found to be

Approximation by Perturbation Techniques

69

and this vanishes because the state is a state of definite parity. Thus for the ground state there is no energy shift in first order perturbation theory. So we have to go to the second order to find any Stark effect on the ground state energy and this will give rise to the so-called quadratic Stark effect. The energy shift in second order is

and the matrix is given as

with the spherical coordinate and where we have used The radial integration above can be done analytically [8], which results in

and thus with

Because this series converges poorly it is worth to find an upper bound with the help of a sum rule.4

This is easy to evaluate, since the ground state wave function is spherically symmetric

and therefore To illustrate degenerate perturbation theory, we calculate the first order Stark effect for the There are four but because the 4 × 4 matrix reduces to a 2 × 2 one

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with eigenvalues Thus the linear Stark effect of the degenerate but not for the

2.

yields a splitting for

1/N-SHIFT EXPANSIONS

The shifted 1/N-expansion yields exact results for the harmonic oscillator and for the Coulomb potential [9]. Because it is simple to derive approximately analytic results for spherically symmetric potentials this approach is useful for power-law potentials the radial coordinate. In this section we will formulate the shifted l / N expansion for arbitrary potentials and derive the energy correction in 4th order perturbation theory. The radial Schrödinger equation in N spatial dimensions is

The essential idea of the 1 / N-shift expansion is to reformulate this Schrödinger equation by using a parameter and a shift parameter a such, that an expansion with respect to those parameters becomes exact for the two limiting cases: Coulomb potential and harmonic potential. This expansion will then be used for a perturbational ansatz to obtain the approximate eigensolutions of the system under consideration. With the ansatz

for the wave function we obtain

which will serve as our starting differential equation. 1st Step. Let us rewrite Eq. (3.55) by shifting the parameter and rescaling the potential

with

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71

By deriving a suitable coordinate shift we will obtain as expansion of our potential a harmonic oscillator potential plus quick converging corrections. To obtain such an expansion it is necessary to remove linear parts with respect to the coordinates in the potential. Therefore we develop the potential around its minimum. 2nd step. Search for a suitable minimum: For large the leading contribution to the energy comes from the effective potential

and this gives

Let us restrict

to those potentials for which

which allows to compute determined.

has a minimum at

in case the position of the minimum

of

thus

is

3rd step. Shifting the origin of the coordinates to In order to shift the origin of the coordinates to the position of the minimum of the effective potential it is convenient to define a new variable x

which yields for the radial Schrödinger equation (3.55)

4th step. Taylor expansion around the effective potential minimum spectively With

re-

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and

the radial Schrödinger equation (3.62) becomes

By reordering this equation with respect to powers in

we obtain

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73

and together with Eq.(3.60) for the terms denoted by (1)

The terms labeled by (2) are independent of and define an oscillator potential

where

which becomes with

5th step. Expansion with respect to orders in For convenience let us introduce the following abbreviations, ordered with respect to powers in

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and thus we obtain for the potential

respectively for the radial Schrödinger equation

with

6th step. Optimized choice of the shift parameter a: The main contribution to the energy comes from

and the next order in energy is given by

Thus we choose a such that this second order contribution vanishes:

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75

7th step. Computation of From Eq.(3.60) we obtain with

Summary of the computational steps. From the implicit equation (3.69) we obtain first the minimum position of and from Eq.(3.60) The oscillator frequency can then be computed via Eq.(3.66) and finally the shift from Eq.(3.68), which completes all necessary steps to obtain the potential expansion. Finally the eigensolutions are computed in Rayleigh Schrödinger perturbation theory. Perturbation expansion. The new potential is equivalent to a shifted harmonic oscillator with anharmonic perturbation. Thus we use the harmonic oscillator eigenfunction and rewrite the anharmonic potential in terms of creation annihilation operators. With the following abbreviation

and introducing the dimensionless new variable

and rescaling the abbreviations defined above via

we obtain after a tedious but straightforward computation the energy in 4th perturbational order:

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Example. is the most important parameter with respect to the speed of convergency in the expansion derived above. As larger as quicker the convergence of the energy for similar potentials. To show this let us discuss briefly the breathing mode of spherical nuclei [10]. Spherical nuclei are 3n-dimensional problems for n nuclei. Thus we will use hyperspherical coordinates, Eq.(2.48). The Hamiltonian is given by

With the product ansatz

and the corresponding Casimir operator

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77

we obtain the radial Hamiltonian in 3n-dimensions

Note, that for we obtain a 3-dimensional problem and the Casimir operator becomes the angular momentum operator and are spherical nuclei, where the figure in front gives the number of nuclei. Because the energy will converge for heavier nuclei much quicker than for light nuclei. Using e.g. Skyrmian or BrinkBoeker forces [10], which are phenomenological nuclei-nuclei forces taking into account spin/isospin corrections, the convergence for the heavier nuclei is much quicker than for the lighter ones. An example is shown in table (3.1).

3.

APPROXIMATIVE SYMMETRY

Exact continuous symmetries for classical systems are related with constants of motion and for quantum systems with conserved quantum numbers, respectively commuting operators. Neither approximative symmetries nor the corresponding approximative invariants are exact symmetries respectively exact invariants. Nevertheless they are useful in the perturbational regime of the system under consideration, because they uncover for weak perturbations the qualitative behavior of the system and simplify considerably the computation. An effective way to derive approximative symmetries is the combination of projection and Lie algebra techniques. The first step is to project the system onto a lower dimensional subsystem in which approximately conserved quantum numbers play the role of parameters and reformulating this subsystem by Liegroup generators. On the next few pages we will analyze this idea [11] by an example: the diamagnetic hydrogen atom in an additional parallel electric field.

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THE DYNAMICAL LIE-ALGEBRA

The dynamical group of a physical system is given by its symmetry group and its spectral generating or transition group. The symmetry represents, e.g., the degeneracy of the quantum system and the existence of constants of motion; the generators of the spectral generating group create the whole quantum spectrum by action onto the ‘ground states’ (the states of lowest weight in representation theory). In this context I used the plural ‘ground states’ because under additional discrete symmetries, it might happen that we can create the whole spectrum only by applying the generating operators on several different states, which are states of lowest energy for one specific discrete symmetry, e.g., the parity. This will become immediately clear by discussing as an example the harmonic oscillator: SU(1,1) is the dynamical group of the harmonic oscillator. Well-known is the special unitary group SU(2), whose generators are given by the angular momentum operator. The corresponding Lie-algebra su(2) is compact, this means physically that the representation consists of a finite basis. In contrast su(l,l) is non-compact and the unitary representation countable but infinite dimensional. A simple example are for su(2) the states for fixed angular momentum which could be half-integer, and the corresponding basis will run from to in steps of one and for su(1,1) the infinite dimensional spectrum of the harmonic oscillator for fixed parity. The corresponding commutator relations are

The generators of SU(1,1) are given by the bilinear products and with â and the boson creation and annihilation operator. Because will increase the oscillator quantum number by two, the multiple action of on the oscillator ground state will create only states with even quantum number and these are the states with positive parity. To create the states with odd oscillator quantum number has to act on the first excited oscillator state which serves as the ‘ground state’ for the odd parity oscillator spectrum. Thus the eigenstates of the harmonic oscillator belong to two different irreducible representations, the even and the odd parity representation spaces. For more details see, e.g., [12], [13] or [14]. SO(4,2;R) is the dynamical group of the hydrogen atom in external fields, where the notation (4,2) reflects the invariant metric. Thus the elements of this

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79

group leave invariant the 6-dimensional space

The operators

with commutator relation

and metric tensor

are the basis of the corresponding Lie algebra, respectively the generators of the group [13]. The symmetry group SO(4;R) of the field-free hydrogen atom is a subgroup of the dynamical group SO(4,2;R). The operators are the angular momentum operators and are the three RungeLenz operators. The most important subgroups of the dynamical group SO(4,2;R) are: –The maximum compact subgroup built by the generators and –The subgroup with generators and for describing the basis in spherical coordinates. – The subgroup useful to describe the hydrogen atom in the oscillator representation. The first and the last subgroup mentioned above will play an important rôle in the following derivation. The equivalence of the corresponding operators projected onto fixed principle quantum number is best uncovered in the boson representation of the hydrogen eigenstates [12].

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and are the annihilation operator for two pair of bosons, spin matrices and

are Pauli

is the metric tensor of SO(2,1;R). In this representation the eigenstates of the hydrogen atom are given by

The principle quantum number is given by

For the SO(4,R) - operators

we obtain the expectation values

and and

3.2

APPROXIMATIVE INVARIANT

In [11] the approximative invariant for the hydrogen atom in strong magnetic fields was derived by the projection and Lie-algebra techniques mentioned above. The first derivation dates back to the classical analysis of [16]. Starting point of the derivation is the Hamiltonian represented by generators projected onto a fixed principle quantum number. Thus the relevant state space is given by the fixed n-multiplet. A perturbation expansion of this Hamiltonian will give us the approximatively invariant operator, hence the generator of a first order perturbational conserved symmetry. The Hamiltonian of the hydrogen atom in external fields in atomic units reads

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with

81

and the magnetic field

pointing into the z-direction, and for parallel electric and magnetic fields. In scaled semiparabolic coordinates

the Hamiltonian

becomes

Because and in parallel electric and magnetic fields are conserved, only the Hamiltonian corrected by the diamagnetic and the electric field contribution are of computational interest. The Zeeman contribution leads to a constant energy shift for fixed and fixed spin. Of course this contribution cannot be neglected in computing spectroscopic quantities like wave lengths and oscillator strengths. A representation of the SO(2,1 ;R) generators is given by

Let us call this operators S.. for the semiparabolic coordinate for Thus the Hamiltonian reads

and T..

The action of this Lie-algebra operators onto the hydrogen eigenstates in the boson representation is given by

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where we have rewritten the standard hydrogenic boson quantum numbers in such a way, that the action onto the principal quantum is uncovered. With the projection

onto a fixed

we obtain

With

and similar for the operators T... In this manner we get from

Expanding this equation with respect to

and in first order

we obtain

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83

The operators above are rather abstract. For obtaining a more physical picture let us rewrite these results in terms of the angular momentum and Runge-Lenz operators. On the n-manifold, and only on the n-manifold, the following isomorphism holds:

By using

and

these finally result in [17]

and thus we obtain for the approximate invariant

Note, this Runge-Lenz operators

are building an ideal, which means

whereas in the standard representation (the non-energy weighted representation) [12] the Runge-Lenz operator are given by

This gives for the approximate invariant

in agreement with the classical representation [16]. Example. For vanishing electric field strength the eigenvalues of the approximate invariant becomes independent of the field strength and thus can be computed by diagonalization. This leads to

with the magnetic quantum number and the spin. The action of the approximate invariant onto the hydrogen eigenstates can be computed easily

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by

The results for the diamagnetic contribution is labeled for some hydrogen eigenstates in Table (3.2), where we have used the field-free notation with an additional indicating that l is no longer a conserved quantum number and thus will mix with all other allowed angular momenta

In Table (3.3) we have compared the results for the state as function of various external field strengths obtained by the approximate invariant and by numerical computations based on the discrete variable and finite elements. These results indicate that even for relatively strong electric and magnetic fields good results can be obtained by using the invariant approximate For the state the results obtained by the approximate invariant are about or even better

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85

1% for magnetic fields up to 5000 Tesla and electric fields of about These limiting field strengths scale roughly with for the magnetic field and for the electric field, the principal quantum number.

4.

TIME-DEPENDENT PERTURBATION THEORY

In this section we will derive an approximate solution to the time dependent Schrödinger equation. The time dependence can enter on two different ways:

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either by a time-dependent perturbation to a zero order time-independent part or by an initial state which is not an eigenstate of a time-independent Hamiltonian (see Chapt.l). An example for the first case is an atom under the influence of a time dependent external electromagnetic field. Even if we are talking about a time-dependent perturbation, this does not mean that we will get a small correction to the energy as we would expect for a time-independent perturbation, because for a time-dependent Hamiltonian the energy will no longer be conserved. Thus the question arises how we could obtain the wave function of the full system from the stationary states of the time-independent system. The time-dependent Schrödinger equation takes the form

and the eigensolutions (with the necessary set of quantum numbers) of the unperturbed system are known. These eigenfunctions serve as a basis for expanding the perturbed wave function

with the time dependent coefficient given by the projection of the wave function onto the n-th basis state

The next step is to apply the time dependent Schrödinger equation onto our wave function expansion and to multiply this equation from the left with a basis state

which results in

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87

This can be rearranged to give

with On the first glance this looks like our searched solution, because Eq.(3.102) is exact, but unfortunately each of the unknown coefficient became a function of all other time-dependent coefficients. A way around that obstacle is in the weak perturbation limit provided by first order perturbation theory, in which we replace all the coefficients on the right-hand-side of Eq. (3.102) by their values at initial time Let us turn on the time dependent perturbation only for a finite time interval

Thus for

and for

with time-independent coefficients Of course, the final state will depend from the initial one, thus we have added the index ‘·, ’ to the coefficients. The probability that the system initially in state is in state for is given by and called transition probability. Let us start with an initially stationary state and a perturbation which has no diagonal elements, thus In first order we use the ansatz which gives

and because of

we obtain finally in first order time-dependent perturbation theory

To gain a better understanding we will discuss two simple cases: the constant and the periodic perturbation.

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Constant perturbation. period

the matrix elements

For a perturbation which is constant during a certain

are time independent. Thus we obtain from Eq.(3.106)

and hence the transition probability becomes (where we will skip in the further discussion the upper index (1))

with

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89

If the energy of the final state equals the energy of the initial state, which is the limit of degenerate states, we get from Eq.(3.108)

and thus the transition probability becomes largest. For respectively the transition probability vanishes in first order, If the interaction time becomes very large compared to the eigentime we obtain for the transition probability with

With

we get immediately

and thus in the limit of an infinite interaction time transitions occur only between degenerate (stationary) states. This result is somewhat surprising, but an infinite interaction time limit with a constant potential is a time-independent conservative system, speaking strictly. Thus we have simply recovered the conservation of energy. Nevertheless we can learn from this result also something about finite interaction times: Significant transitions for a finite interaction time will only occur for small energy differences between the initial and final state under consideration. Thus only those states will be populated significantly for which For the transition energy we obtain directly

Note, that the right-hand-side of this equation is constant. Thus as smaller the interaction time as larger the transition energy. Eq.(3.113) represents the energy-time uncertainty relation. It is of importance that this uncertainty differs in its interpretation significantly from the Heisenberg uncertainty For the Heisenberg uncertainty the coordinates and momenta are taken at the exactly same time for the same state of the system under consideration and the Heisenberg uncertainty relation tells that both cannot be measured together exactly. Each energy can be measured exactly at any time. is the

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difference between two energies measured at different times, but this does not imply that one of the energies of one of the stationary states under consideration possess an uncertainty of the energy at a certain time. The differential transition probability becomes for long interaction times in first oder time-dependent perturbation theory time-independent and proportional to the squared interaction potential due to Eq.(3.111)

Usually final and initial states are not isolated bound states but states in the continuum. Therefore speaking practically the transition probability is the probability to populate all states sufficiently close in energy to the final state Of course in general is multi-valued quantum number and thus all other quantum numbers (e.g. angular momentum l and magnetic quantum number which do not refer to the energy have to be considered in addition. It depends on the special physical situation under consideration if only those states have to be taken into account which have identical additional quantum numbers or if one have to sum over all or some degenerate quantum numbers. Let be the density of final states. Then the total differential transition probability is given by

and this equation is called Fermi’s Golden Rule. Periodic perturbation. bation

Let us consider finally the case of a periodic pertur-

where ± refers to the sign in the exponential function. Again, initially the system will be in the n-th stationary state. In first order time-dependent perturbation theory, we obtain from Eq.(3.106)

and thus

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91

Similar to the discussion above we obtain for large interaction time

and thus for the transition probability

and for the differential transition probability

Due to this periodic perturbation those states

are populated for which

Hence if we define again the density of final states in the continuum (see discussion above) by the (total) differential transition probability becomes

The ±-sign differs two situations: For the +-sign states with energy are populated. Thus by the transition the final states are lower in energy which means the system looses the energy This happens for stimulated emission in which the system due to the external periodic perturbation radiates. Stimulated emission is exemplified by the output of lasers and is the exact opposite of absorption. In the case of absorption the system gains the energy by the transition from and thus the final states will be higher in energy and this is exactly the situation which happens by absorption of light, labeled above by the –-sign.

Notes 1

is the magnetic induction measured in atomic units . In addition to the Zeeman contribution the diamagnetic contribution becomes important for strong magnetic fields and/or highly excited states, the so-called Rydberg states. 2 The quantum number is still conserved 3 We will not consider the coupling to the continuum due to tunneling below the classical ionization limit. 4 The relation

is called a sum rule.

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5 The subgroup SO(2,1;R) [15] is generated by the operators

which leave the angular momentum quantum numbers invariant and shift the principle quantum number by one.

References [1] Cohen-Tannoudji, C., Diu, B. and Laloë, F. (1977). Quantum Mechanics II John Wiley & Sons, New York [2] Flügge, S. (1994). Practical Quantum mechanics (2nd printing) SpringerVerlag Berlin [3]

J. and Vrscay, E. R., (1982). “Large order perturbation theory in the context of atomic and molecular physics - Interdisciplinay aspects”, Int. Journ. Quant. Chem. XXI, 27–68

[4] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes Cambridge University Press Cambridge [5] Schweizer, W. and Faßbinder, P., (1997). “Discrete variable method for nonintegrable quantum systems”, Comp. in Phys. 11, 641–646 [6] Bohigas, O., Giannoni, M.-J. and Schmit, C. (1986). in Lecture Notes in Physics Vol. 262, edts. Albeverio, S., Casati, G. and Merlini, D. SpringerVerlag Berlin [7] Reichl, L. E. (1992). The transition to chaos Springer Verlag, New York [8] Bethe, H. A. and Salpeter E. E. (1957). Quantum Mechanics of One- and Two-Electron Atoms Academic Press, New York [9] Imbo, T., Pagnamenta, A., and Sukhatme U., (1984), “Energy eigenstates of spherically symmetric potentials using the shifted l / N expansion”, Phys. Rev. D 29, 1669–1681

References

93

[10] Ring, P. and Schuck, P., (1980). The nuclear many body problem Springer, Heidelberg [11] Delande, D., Gay, J. C., (1984). J. Phys. B17, L335 – 337 [12] Wybourne, B. G., (1974). Classical Groups for Physicists John Wiley & Sons, New York [13] Barut, A. O., (1972). Dynamical Groups and Generalized Symmetries in Quantum Theory Univ. of Canterbury Pub., Christchurch, NewZealand [14] Kleiner, H., (1968). Group Dynamics of the Hydrogen Atom in Lect. Theor. Physics XB, eds. Barut, A. O., and Brittin, W. E., Gordon & Breach, New York [15] Bednar, M., (1977) “Calculation of infinite series representing high-order energy corrections for nonrelativistic hydrogen atoms in external fields”, Phys. Rev. A15, 27–34 [16] Solov’ev, E. A., (1981) “Approximate motion integral for a hydrogen atom in a magntic field”, Sov. Phys. JETP Lett 34, 265 – 268 [17] Bivona, S., Schweizer, W., O’Mahony, P. F., and Tayler, K. T., (1988) “Wavefunction localization for the hydrogen atom in parallel electric and magnetic fields”, J. Phys. B21, L617

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Chapter 4 APPROXIMATION TECHNIQUES

As a matter of fact, even the simplest system, the hydrogen atom, represents a system of two particles. Thus in this chapter we will concentrate on techniques suitable for systems which cannot be reduced to a single particle problem. Of course this does not mean that, e.g., finite element techniques or perturbation techniques are only useful for single particle problems. But in most practical computations different techniques will be used in combination, e.g., one- or two-dimensional finite elements as an ansatz for a part of the higher dimensional wave function. In the first section we discuss the variational principle followed by the Hartree- Fock method. Simplified the Hartree-Fock technique is based on a variational method in which the many fermion wave functions are a product of antisymmetrized single particle wave functions. This ansatz leads to an effective one-particle Schrödinger equation with a potential given by the selfconsistent field of all other fermions together. The density functional theory is an exact theory based on the charge density of the system. Density functional theory includes all exchange and correlation effects and will be discussed in section three. Section four is devoted to the virial theorem, which plays an important role in molecular physics and the last chapter to quantum Monte Carlo methods. With the increasing power of computer facilities quantum Monte Carlo techniques provide a practical method for solving the many body Schrödinger equation.

1.

THE VARIATIONAL PRINCIPLE

In understanding classical systems variational principles play an important role and are in addition computationally useful. Thus two questions occur: Quantum systems are governed by the Schrodinger equation. Is it possible to formulate a quantum variational principle? If we have found such a variational 95

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principle, will it enable us to solve the Schrödinger equation in such situations where we found neither an exact solution nor a perturbational treatment was amenable? In the following we will show, that we are able to replace the Schrödinger equation by a variational method - the Rayleigh-Ritz method. Even if the Rayleigh-Ritz method in its naive numerical application seems to be rather restricted, many other numerical techniques, e.g. the finite element method, are justified by the equivalence of the Schrödinger equation to a variational principle,

1.1

THE RAYLEIGH-RITZ VARIATIONAL CALCULUS

The variational method is based on the simple fact that for conservative systems the Hamiltonian expectation value in an arbitrary state has to be greater or equal the ground state energy of the system. Let us assume, for simplification, that the Hamiltonian spectrum is discrete and non-degenerate. Eigenstates hold

and each wave function Hilbert space basis. Thus

can be expanded with respect of this complete

which proves the contention above. The Ritz Theorem. The Hamiltonian expectation value becomes stationary in the neighborhood of its discrete eigenstate:

Proof:

Approximation Techniques

For any state

97

is a real number. Thus

Let then

Because this holds for any varied ket, it is also true for

and thus becomes stationary if and only if is an eigenstate. Therefore the ‘art of variationing’ is to chose judiciously a family of trial wave function, , that contains free parameters The variational approximation is then obtained by

1.1.1 EXAMPLES To illustrate the utility of the variational procedure we will discuss two examples, the harmonic oscillator and the helium atom. The harmonic Oscillator.

The Hamiltonian of the harmonic oscillator is

and the trial wave function is given by

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Thus

and therefore we obtain

and with Eq.(4.5)

which is the correct ground state energy and wave function of the harmonic oscillator. With the ansatz we would have obtained the correct first excited state, which is the state of lowest energy with negative parity. The calculations above are only on pedagogical use to familiarize ourselves with the variational technique. Because the trial functions chosen included already the exact wave function, we are not able to judge by this example its effectiveness as a method of approximation. Therefore we will now discuss a second example. The helium atom. The helium atom is composed of two electrons in the s-shell. If the nucleus is placed in the origin, and if the electron coordinates are labeled and then the Hamiltonian is (in atomic units)

with Since we are interested in the ground state, both electrons must occupy the lowest energy state. Therefore the simplest variational ansatz is the product of two hydrogen-like ground state eigenfunctions

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99

Because we are concerned with a two electron system, hence a fermion system, the total wave function has to fulfill the Pauli principle. Thus we have to antisymmetrize the trial function against exchange of the two electrons. Because the spatial part is symmetric the spin part has to be antisymmetric. Coupling the two electron spins leads to the triplet S = 1 state being the symmetric and the singlet S = 0 state the antisymmetric state under exchange of the two electron spin coordinates. Therefore the total correct trial function for the ground state is given by With

we obtain

and thus the energy becomes after some elementary integration steps

Minimizing Eq.(4.11) as a function of

leads to

and The computation above holds for any two electron system. The experimental measured quantity is the ionization potential I given by where is the ground state energy of the hydrogen-like single ionized atom. Thus equals respectively

Table (4.1) lists the variational and the measured values for some two-electron systems. Surprisingly the difference between the variational and the experimental result is approximately invariant, at least for the first four entries. This is an effect of the electron-electron interaction, which becomes for higher and higher ionized atoms relatively less important because the nucleus/electron attraction becomes much stronger. Higher approximation in the variational treatment take the electron-electron interaction explicitly into account. Hylleraas suggested as trial function

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With this ansatz Pekeris [1] obtained and the experimental result is And this exhibits one of the strengths of variational methods: In many cases they allow to obtain highly accurate values for single selected states, which allow to study, e.g., small relativistic atomic effects.

1.2

DIAGONALIZATION VERSUS VARIATION

1.2.1 DIAGONALIZATION AS VARIATIONAL METHOD In the variational computation the bound states of the Hamiltonian are found within a subspace of the Hilbert space. This subspace is given by the variational parameters in the examples above. Of course this set of trial functions could also be expanded by a set of basic vectors of the complete Hilbert space basis. Therefore we could reinterpret the variational calculus above and equivalently use a finite set of N Hilbert space basis vectors which span the relevant submanifold. For a trial state

the Hamiltonian expectation value is given by

with For a orthonormal Hilbert subspace basis the elements of the overlap matrix are

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101

For the Hamiltonian bound states the derivative with respect to the of the expectation value vanishes, which leads to the linear system of equations

Eq.(4.16) is a generalized eigenvalue equation which can be written in matrix notation Thus diagonalization procedures based on a Hilbert subspace are equivalent to a variational ansatz. Variational results, for both ground state and excited states, are always superior to the correct eigenvalue. Including more basis functions into our set, the Hilbert subspace becomes larger. Consequently the computational results will converge towards the exact ones by increasing the basis size. A very effective method solving large scale eigenvalue equations is given by Arnoldi and Lanczos methods [2] [3]. In the following subchapter we will discuss the Lanczos method under a slightly different point of view. 1.2.2 THE LANCZOS METHOD Spectrum Transformation Techniques. Standard diagonalization techniques like Householder's method scale with if is the dimension of the matrix. Lanczos methods are usually used to compute a few eigenvalues of a sparse matrix and scale with Therefore Lanczos techniques becomes efficient for large eigenvalue problems. Because the Lanczos method converges quickly only for extremal eigenvalues, spectrum transformation techniques [2] are used to compute non-extremal interior eigenvalues of the complete spectrum. Therefore speaking practically, if one needs to compute a large set of eigenvalues of the complete spectrum the total set is windowed into smaller subsets. Each of these subsets are spectrum transformed, to map its interior eigenvalues onto a new spectrum for which the new transformed eigenvalues become extremal. The essential idea is very simple. Let us assume the subset of eigenvalues we are interested in, is covered by the interval and is an element of this interval but not an eigenvalue. Instead of computing the eigenvalues of our original matrix we derive a new matrix with a shifted and inverted spectrum, such that this spectrum becomes extremal around the selected value

Let be our original generalized eigenvalue problem with positive. Thus we can map the generalized problem onto an ordinary one with shifted eigenvalues

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which we can rewrite as

and therefore we arrive at

Thus the eigenvalues around are mapped onto new eigenvalues becomes extremal. The original eigenvalues are given by

which

The Lanczos algorithm. Let the Hamiltonian of the system under consideration and a randomly selected normalized state. By the Lanczos procedure we will derive a tridiagonal Hamiltonian matrix, whose eigensolutions will converge towards the exact ones. 1st step: We compute a new state by applying the Hamiltonian onto our initial state. Because the initial state was randomly chosen, it will not be an eigenstate of Hamiltonian and thus and

are not parallel. By Gram-Schmidt orthonormalization

we obtain a state orthonormal to our randomly chosen first state. The Hamiltonian matrix elements are

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103

Note, by using the numerical results from the Gram-Schmidt orthonormalization the total computation could be reduced significantly, because already all matrix elements are computed by the step before. 2nd step: We compute the next state again by applying the Hamiltonian onto our previously derived state:

and by Gram-Schmidt orthonormalization

The new matrix elements are

The elements vanishes because can be written as a linear combination of the two states which are by construction orthonormal to Similar holds for all non-tridiagonal elements and thus

Due to the construction states with state labels which differs by more than two are already orthonormal and thus Gram-Schmidt orthonormalization runs only over the two predecessors. The general construction is

and by Gram-Schmidt orthonormalization

Continuing this process leads finally to a tridiagonal matrix whose eigenvalues converge to the Hamiltonian ones. Based is this construction on the Krylov space [3]. The Krylov space is spanned by all basis vectors generated by multiply applying the Hamiltonian onto the first randomly selected one, The eigenvalues and eigenvectors of tridiagonal matrices are efficiently calculated by QR-decomposition. The QR-decomposition is based on the following construction: Let our tridiagonal Lanczos Hamiltonian matrix. Then

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we decompose this matrix in a upper triangular

and an orthogonal matrix

Commuting this decomposition results in a new matrix and a new decomposition Continuing this process leads to

and hence all matrices have exactly the same eigensolutions and converges towards a diagonal matrix. For general symmetric matrices the QR-decomposition converges slowly, but is an efficient algorithm for tridiagonal matrices. Routines can be found, e.g., in [4].

2.

THE HARTREE-FOCK METHOD

In this section we will discus briefly the Hartree-Fock method. The HartreeFock theory provides a simplification of the full problem of many electrons moving in a potential field. The essential idea is to retain the simplicity of a independent single-particle picture. This means that in the spirit of the Ritz variational principle we search for the ‘best’ many particle eigenstate approximated by a product of single particle states. The Hamiltonian for a Z electron system (atom or ion) is given by

with eigenstates (The Hamiltonian above is already approximative because we have neglected the atomic nuclei.) In a first step we approximate the Z-electron wave function by a product state

Physically this ansatz implies that each electron in the multi electron system is described by its own wavefunction. This implies that each single electron experiences an equivalent potential due to all other electrons and the nucleus.

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Because our wave function is normalized all single electron wave functions are also normalized. Therefore the Hamiltonian expectation value is given by

Variation with respect to single particle states leads to

under the additional constraint of orthonormal

respectively

This integro-differential equation is called Hartree equation. The integral potential term, the effect of all other particles on each particle, is a selfconsistent potential and the idea approximating a physical system by a selfconsistent potential the basis of the self-consistent field method [4]. Eq.(4.30) describes the single electron moving in the Coulomb potential plus a potential generated by all other electrons. However, the self-consistent potential

depends on the charge density which we know only after solving Eq. (4.30). The total energy E is given by the Hamiltonian expectation value (4.29). The expectation value of the Hartree equation (4.30) counts the repulsive electronic interaction twice, hence the correct total energy is not obtained by the sum of the single particle energies but by

In the Hartree solution above, we have not taken into account that electrons are fermions and thus the total wave function have to be antisymmetric. The

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antisymmetrization of a product of N single particle states can be obtained with the aid of the antisymmetrizer

with the permutation operator and the sum extends over all possible permutations, where is +1 if the permutation is even, otherwise –1. It is more convenient to write the antisymmetric wave function of orthonormized single particle states by a Slater determinant

where includes spin and space coordinates. This means that we search in the spirit of the self-consistent field theory for the ‘best’ Slater determinant. Thus the Pauli principle is taken into account, and we get from the variational procedure above the Hartree-Fock equation, which differs from the Hartree equation (4.30) by the appearance of an additional exchange-potential term. It can be verified easily, that the Slater determinant is normalized. We obtain for the one electron part of the Hamiltonian

and the electron-electron interaction

therefore the total energy is given by

For more details see, e.g., [5].

3.

DENSITY FUNCTIONAL THEORY

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In the Hartree-Fock formalism the many body ansatz for the wave function is written in the form of a Slater determinant and the solution obtained in a selfconsistent way. Similar to the Hartree idea an effective independent particle Hamiltonian is derived here. The solution are then self-consistent in the density

where the sum runs over the N spin-orbitals single particle Hamiltonian

lowest in energy of the quasi

N being the number of electrons in the system. The first three terms are the kinetic energy, the electrostatic interaction between the electron and the nuclei and the self-consistent potential. Therefore this equation differs only in the exchange and correlation potential from the corresponding Hartree-Fock equation. The total energy E of the many-electron system is given by

with

defined by the functional derivative

These equations were first derived by Kohn-Sham [6]. For more details see, e.g., [7]. For applying the density functional method the ground state energy as a function of density have to be known. The difference to the Hartree-Fock theory above is the replacement of the Hartree-Fock potential by the exchangecorrelation term, which is a function of the local density. The exact form of the exchange-correlation functional is unknown. The simplest approximation is the local density approximation (LDA). In the local density approximation the single particle exchange-correlation energy is taken local at the position approximated by a homogeneous electron gas and

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For a homogeneous electron gas the single particle exchange-correlation energy can be separated in the exchange and correlation part

Rewriting the electron density by the Wigner-Seitz radius

we obtain

and

and by Monte Carlo computations [8]

Hence we are able to compute the exchange-correlation energy and potential term

Of course the local density approximation is only a crude approximation, which can be further improved by gradient terms in the density, pair correlation terms and so forth. For more details see [9].

4.

THE VIRIAL THEOREM

The virial theorem provides exact relations between the energy and the expectation value of the kinetic and potential energy operator. In this section we will derive the virial theorem, discuss its physical consequence and its numerical usefulness.

4.1

THE EULER AND THE HELLMAN-FEYNMAN THEOREM

The Euler Theorem. of degree s if

A function

is called homogeneous

For example, the harmonic oscillator in n dimensions is homogenous of degree "2", or the two-particle Coulomb potential is homogeneous of degree " –1".

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Euler’s theorem states that any function f which is homogeneous of degree s holds

The Hellman-Feynman Theorem. depends on a real parameter and

Let

be an hermitian operator which

with The Hellman-Feynman theorem states that

This relation can be derived easily: By definition we have

If we differentiate this relation with respect to

which proofs the theorem above. For the mean value of the commutator of Hamiltonian eigenstate we obtain immediately

since

we obtain

with any operator  in an

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4.2

THE VIRIAL THEOREM: THEORY AND EXAMPLES

4.2.1 DERIVATION In classical mechanics the derivation of the virial theorem is based on the time average of the relevant phase space functions. In quantum dynamics we mimic this ansatz by studying the expectation or mean values. Let (or in general with twice differentiable and

From

and the Schrödinger equation

we obtain

and with Eq. (4.49)

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hence

For

we obtain the virial theorem:

with the kinetic energy operator. Using Eq.(4.5 1b) we find for the harmonic oscillator and for the (1-dimensional) hydrogen atom

Setting

leads to the hypervirial theorem

useful in perturbation expansions. 4.2.2

THE VIRIAL THEOREM APPLIED TO MOLECULAR SYSTEMS Consider an arbitrary molecule composed of nuclei and q electrons. We shall denote by the classical position of the nuclei, and by and the position and momenta of the electrons. We shall use the Born-Oppenheimer approximation here, considering the position of the nuclei as given nondynamical parameters. The Born-Oppenheimer approximation is based on the great differences between the masses of the electrons and nuclei in a molecule, which allows to consider the nuclei motion as frozen compared to the electronic motion. Therefore only the electronic coordinates and momenta are considered as quantum operators. In Born-Oppenheimer approximation the Hamiltonian is given by

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with

where the first two potential terms are quantum operators due to the electron interaction and the only effect of the last term is to shift all energies equally. The second potential operator depends parametrically on the position of the nuclei. This position is derived by minimizing the total energy. The electronic Hamiltonian is given by and depends parametrically on the nuclei position with eigenenergies and wavefunctions Due to the Euler theorem the potential is homogeneous of order –1 and due to the Hellman-Feynman theorem the Hamiltonian holds To derive the virial theorem for molecules we start by computing this commutator.

Therefore

because due to the Born-Oppenheimer approximation depends only parametrically via the potential term from the nuclei’s position. Application of the Hellman-Feynman theorem results in

and because obviously

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we find for the virial theorem applied to molecules

By this simple result we are able to compute the average kinetic and potential energies from the variation of the total energy as a function of the position of the nuclei. In analogy to the derivation above we can also derive the virial result for the electronic potential energy

(Of course due to the Born-Oppenheimer approximation the total kinetic energy is exactly the kinetic energy of the electronic motion.) 4.2.3 EXAMPLE For diatomic molecules the energy depends only parametrically from the diatomic distance and we obtain from Eq.(4.60a,4.60b)

because

is homogeneous of order 1 and thus

Loosely speaking, the chemical bond is due to a lowering of the electronic potential energy as a function of the nuclear distance. Let be the total energy of the system when the nuclei are infinitely far apart from each other. To build a stable molecule or molecule-ion the total energy have to have a minimum at a finite nuclear distance Thus due to the virial theorem

and at infinite distance

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Therefore stability of the molecular system necessitates

The molecule ion As an application of the virial theorem we will discuss the molecule-ion. The coordinates are shown in Fig.(2.3) and the Hamiltonian is given by

where the nuclear distance serves as parameter. In the limit "electron close to the first nucleus, second nucleus far away" respectively "electron close to the second nucleus, first nucleus far away" the total energy of the molecule-ion is approximately given by the hydrogen ground state energy. Therefore we chose as test states a linear combination of hydrogen-like ground states

with

for hydrogen. Therefore the variational task is to compute the optimized nuclear distance of lowest energy for the selected test state. Therefore we have to calculate the stationary state of

From we obtain

and non-trivial solution for

The wavefunctions are normalized and real

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The following calculation are simplified by using elliptic coordinates (2.41), with For the overlap S we obtain

For symmetry reasons the Hamiltonian matrix holds

with – the hydrogen ground state energy and C the Coulomb integral. The Coulomb integral can be calculated easily in elliptic coordinates

and thus

Note

and hence, as expected, the energy equals the hydrogen ground state energy for infinitely distant nuclei.

with the exchange integral and hence

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To compute the energy eigenvalues we have to solve the energy determinant which is given by

using the abbreviations

this results in

which holds Hence the energy becomes the hydrogen ground state energy for infinitely distant nuclei and denotes the bonding and the antibonding case. The eigenstates are given by

where the ground state (bond state) is given by , Eq.(4.78b), and symmetric under particle exchange. In Fig.(4.1) the Coulomb, exchange and overlap integral are plotted. For infinitely distant nuclei the overlap and exchange integral goes to zero. This is obviously because the system becomes composed of a hydrogen atom and an additional infinitely distant proton and the interaction between both becomes arbitrary small. In the infinite limit the Coulomb integral converges to the Coulomb potential and thus goes like It is instructive to calculate the potential and kinetic energy directly. But note these direct computation will lead to wrong results! The potential expectation value is given by

and the kinetic energy by

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Therefore for the kinetic energy difference we obtain

Thus by direct computation the kinetic energy difference becomes smaller zero because the exchange integral is smaller than the overlap integral, see Fig.(4.1). But this result is in contrast to our statement above, that for binding states the kinetic energy will increase by bringing the two components from infinity closer together. Therefore, in general, variational results might lead to wrong values for special operators. The variational results guarantee only to minimize the energy with respect to the class of test states. The only way to compute the correct kinetic or potential contribution is given by the virial theorem. From the virial theorem we obtain

and thus the correct behavior of the mean value of the kinetic and the potential energy operator.

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5.

QUANTUM MONTE CARLO METHODS

Monte Carlo methods are based on using random numbers to perform numerical integration. The term quantum Monte Carlo does not refer to a particular numerical technique but rather to any method using Monte Carlo type integrations. Thus the question is for which systems are Monte Carlo methods effective? To answer this question, let us first consider standard numerical integration based on an order-k algorithm, with an error of the order for a d-dimensional integral. Let us assume for simplicity that the integration volume is a hypercube with length l. The step size of our conventional numerical integration method is thus the hypercube contains integration points and therefore the error scales as1 The Monte Carlo integration in one dimension converges very slow, but this error is independent from the dimension. Thus we obtain the same scaling behavior for and hence Monte Carlo integration becomes more efficient than the selected conventional order-k algorithm when Therefore quantum Monte Carlo methods are efficient for solving the few- or many-body Schrödinger equation.

5.1

MONTE CARLO INTEGRATION

5.1.1 THE BASIC MONTE CARLO METHOD One of the most impressive Monte Carlo simulations is the experimental computation of the number which was presented at the exhibition “phenomena” in Zurich, Switzerland, 1984. Suppose there is a square with length and an interior circular hole with radius Each visitor of the “phenomena” was advised to throw a ball into the direction of the square without aiming at the hole. Thus each visitor is a random number generator. The surface of the square, is and the surface of the interior hole and hence the probability p to hit the hole is

Counting the number of balls flying through the hole will give us (a few thousand visitors later) a good approximation of the number Suppose we want to calculate the integral of an integrable function f over the real interval [a, b]

Standard methods are the trapezoidal rule [4]

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Where we have used N equally spaced interior intervals of width and applied the trapeze rule to each of the intervals. The trapezoidal rule is exact for linear polynomials. A very efficient method consists of repeating the integration technique based on Eq.(4.82) for successive values of each having half the size of the previous one. This gives a sequence of approximations, which can be fitted to a polynomial and the value for this polynomial at yield a very accurate approximation to the exact result. This accurate quadrature is called Romberg integration. Suppose we want to integrate a function f over a region with complicated shape. Remember the experiment above. Having this idea in mind it should be no problem: The first step will be to find a simpler region which contains as subarea. Creating a random sample uniformly distributed in we have to decide if lies inside or outside the integration area If it lies inside it will count, if it is outside it does not contribute. Therefore we obtain

with otherwise zero. The basic theorem of the Monte Carlo integration [10] of a function / over a multidimensional volume V is

with the expectations value of a function over sample points

defined by its arithmetic mean

5.1.2 THE METROPOLIS STRATEGY The discussion above shows the Monte Carlo integration involves two basic operations: Generating randomly distributed vectors over the integration volume, and evaluating the function at this values. While the second task is straightforward, the generation of uniformly distributed random numbers is not obvious. A quick and dirty method [4] is given by the recurrence relation

where is called the modulus and a and are positive integers. If a, c and are properly chosen, Eq. (4.84) will create a sequence with period where the initial value is called seed. In this case, all integers between 0 and will occur.

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Suppose we have generated an uniformly distributed sequence of random numbers. How can we obtain from this sequence a different probability distribution? Let and be the probability distribution of respectively From

we obtain obviously

Therefore by selecting a suitable transformation function we can compute other probability distributions from the ‘quick and dirty’ method mentioned above. As an example let us assume that For an uniformly distributed sequence the probability is which leads immediately to a Poisson distribution:

But note, the sequence of numbers created above are not really random distributed. The computer is a deterministic machine and thus most methods create only a quasi random number sequence of finite length. Now let us return to Eq.(4.84). Eq.(4.84) states that the integral over a d-dimensional scalar function / can be approximated by

where we have scaled the coordinates such that V = 1, The uncertainty in the Monte Carlo quadrature is proportional to the variance of the integrand. By introducing a positive scalar weight function

we can rewrite the integral above

and change the variable from

hence the integral

so that the Jacobi-determinant becomes

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The potential benefit of the change of variables is to select the scalar weight function such that the squared variance

will be minimized and thus the Monte Carlo integration optimized. If we choose that it behaves qualitatively like the variance will be reduced and the approximation improved. This method is called important sampling Monte Carlo integration. Now let us go one step further. The Monte Carlo quadrature can be carried out using any set of sample points. But obviously the efficiency will depend on the selected probability distribution. An uniform random distribution will collect values of the function in areas where the function nearly vanishes as well as in areas where the function dominates the sum. Thus the convergence will be poor. The transformation above leads to a sample which concentrates the points where the function being integrated is large. The essential idea of the Metropolis algorithm is to select a random path in space such that the convergency is improved, which means to avoid those areas where the function does not noticeable contribute to the total sum. Suppose that the current position on the random walk is and that the random move would lead to the new position

Each of these positions in space are associated with a certain probability The move from a position to the position is performed by a walker. If the move is accepted the walker will move from to otherwise the walker remains where it is and the next trial step will be selected. In this way it will be possible for the walker to explore the configuration space of the problem. Thus the only remaining task will be a rule how to decide when the walker will move and when the walker will stay and take the next trial. Let us denote by the probability to move from to . In the Metropolis algorithm the probability of accepting a random move is given by

Thus if is smaller a certain number in the range [0,1], either arbitrarily selected or chosen randomly at the beginning of the walk, the move will be accepted, otherwise rejected. Therefore graphically the walker will mainly move uphill. Practically the computation starts by putting N walkers at

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random positions, then select a walker at random. This walker will be shifted to a new position and the Metropolis criteria will be calculated to decide if the move will be accepted or rejected. This procedure will be repeated until all walkers moved through their individual random path. Therefore the summation runs over many walkers and their individual path. This technique guarantees that a selected random path will not mainly explore the region for which the function vanishes nearly and on the other side the selection of many random walkers guarantees that the whole area of interest will be explored.

5.2

THE VARIATIONAL QUANTUM MONTE CARLO METHOD

In chapter 3.5 we studied the variational calculus to obtain the Hamiltonian eigensolutions. The essential idea of the Rayleigh Ritz variational principle was to parameterize the wave function and to compute the minimum of the energy as a function of the non-linear parameter. The necessary computation of Hamiltonian expectation values is based on the evaluation of integrals. The variational quantum Monte Carlo method is a combination of the variational ansatz and the Monte Carlo integration. First step of the variational ansatz is to choose a trial wave function with parameters and to compute the energy expectation value

With the energy expectation value above can be rewritten as

with

the local energy, and

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the stationary distribution. The procedure is now as follows. The Metropolis algorithm is used to sample a series of points, in configuration space. The path is selected such that the fraction is

At each points of these paths the local energy, the mean

is evaluated and finally

computed. To optimize the computation a large collection of walkers is used. A generalization to obtain also excited states via diffusion quantum Monte Carlo can be found in [11] and an application to atomic systems in [12].

5.3

THE DIFFUSION QUANTUM MONTE CARLO

A further quantum Monte Carlo method is the so-called diffusion quantum Monte Carlo method. The idea is to interpret the time-dependent Schrödinger equation (1.3) as a diffusion equation with potential, by using its imaginary time form

Expanding obtain

with respect to the Hamiltonian eigensolution,

we

and for the imaginary-time evolution

In the long-time limit this state will converge to the state of lowest energy in the wave function expansion. Thus as long as is not orthogonal to the Hamiltonian ground state we obtain

Introducing an energy shift

leads to

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with the total potential. Note, for n-body systems as well as will be a 3 . dimensional vector. can be reinterpreted as a rate term describing the branching processes. The energy shift is an arbitrary parameter that effects only the unobservable phase of the wave function but not its modulus and can be understood as a trial energy. If the ground state energy, will grow exponentially in time and for decrease exponentially. Thus solving Eq.(4.95) for various values of and monitoring the temporal behavior of could help us to optimize the energy shift If remains stationary we would have found the correct energy and the correct ground state wave function. A computer calculation could be set up in the following way: An initial ensemble of diffusing particles representing is constructed. The imaginary-time evolution is accomplished by considering a snapshot after a imaginary-time step The entire equation is then simulated by a combination of diffusion and branching processes. For most problems this algorithm is not satisfactory and a very inefficient way to solve the Schrödinger equation, because whenever the potential becomes very negative, the branching process blows up. This leads to large fluctuations in the number of diffusing particles and thus to a large variance and hence an inaccurate estimate of the energy. As already discussed above, important sampling will weaken the variance and hence optimize the approximation. Thus the first step is to find a weight function to optimize the Monte Carlo steps. This will be achieved with the help of a guiding function. For this purpose a trial approximation, to the ground state wave function is introduced and a new distribution

defined. The trial wave function, could be derived, e.g., from a HartreeFock calculation. Inserting Eq.(4.96) in Eq.(4.93) yields a Fokker-Planck type of equation

with

the so-called quantum force, and

the branching term, which gives rise to the important guiding function

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What do we gain by this method? If would be an exact eigenvalue would become independent of Thus if is a reasonable approximation to the ground state would be sufficiently flat. Guide functions were first introduced by Kalos et al. [13]. Equation (4.97) is a drift-diffusion equation for The branching term is proportional to which for close to a Hamiltonian eigenstate need not become singular when does. Note, for a true Hamiltonian eigenstate the variance in would vanish. The procedure outlined here might fail in some cases. The distribution of walkers can only represent a density which is positive everywhere. The ground state of a boson system is everywhere positive, but this would not be true for fermion systems. This leads to the so-called fermion problem. Besides reducing the fluctuations in the number of diffusing particles has an additional important role for fermionic systems: determines the position of the nodes of the final wave function. Thus the accuracy of the initial nodes due to determines how good the estimate of the ground state energy, will be. This can be understood easily by considering the long imaginary-time limit

Thus have to vanish at the nodes of to obtain the true fermionic ground state in the imaginary time limit. This approximation is called the fixed-node method. Normally the positions of the zeros are only approximately known. In these cases the nodes are estimated and empirically varied until a minimum of the energy is found. This advanced variant is called released-node method [11]. The concrete algorithm runs as follows: We start with an initial set of random walkers distributed according to and let the walkers diffuse for a imaginary time step with a certain transition probability given by

with energy shift

the approximate Green function in the limit of a vanishing

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In order to deal with the nodes in the fermion ground state we must use the fixed-node or released-node approximation. A trial displacement is given by

with a 3 . -dimensional vector for particles and 3 space dimensions, a 3 . -dimensional displacement vector, the time step for which the particle diffuse independently. The three-dimensional subspace displacement vector for each particle is a Gaussian random variable with mean of zero and a variance This trial displacement is accepted with probability

Then branching is performed according to the branching probability

Next, produce copies of the walker, with a uniform random number. The steps described above should be repeated several times before computing the mean of and the new trial energy from

This procedure is then repeated until there is no longer a detectable trend in the mean of At this point the steady-state has been reached. More details about quantum Monte Carlo methods in general can be found in [14] and a brief and graphic presentation in [15].

5.4

THE PATH INTEGRAL QUANTUM MONTE CARLO

Fundamental path integral concepts. Instead of describing quantum systems by its Schrödinger equation there is an alternative of quantizing a classical system via the Feynman path integral, which is equivalent to Schwinger’s action principle. The first step on the way to quantizing a system is to rewrite the problem in Lagrangian form. Assume that we have prepared the wave function at time and position and we want to observe the wave function at later time and position Both wave functions are connected according to

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where the integral kernel follows the time dependent Schrödinger equation

Thus the integral kernel is a retarded propagator which describes the preparation of the quantum system at and its observation at Due to Feynman this propagator can be obtained by

with the classical Lagrangian. For causality reasons the timedifference always has to be positive definite. The propagator K fulfills the important Kolmogoroff-Smoluchovski or folding property

Thus we could introduce an infinite number of arbitrary small intermediate steps and approximate the propagator piecewise in each intermediate step by the free particle Lagrangian of the same total energy, due to Eq.(4.105). Note, that the integration in Eq.(4.106) runs over all possible space vectors Thus after putting all pieces together all paths, not only the classically allowed one, will contribute to the Feynman path integral. In the semiclassical limit these paths will reduce to the classical allowed one. It turned out that the path integral formalism becomes important especially in studying quantum chaos or quantum systems which can only be described by density operators of mixed states. An example are quantum systems which are in contact with a heat bath of temperature T > 0. Due to the importance of paths in the Feynman formalism it is straightforward to try to merge path integral ideas with Monte Carlo methods. For more details about Feyman path integrals see [16] and for semiclassical applications [17]. For quantum systems which are in contact with a heat bath of temperature the density matrix in the coordinate representation (see Chapt. 1.5.1) is given by

with

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The expectation value of an observable  is given by Evidently is the canonical partition function. If we would know the quantum density function, the road would be free for a Monte Carlo simulation in the spirit of the classical case. However, the explicit form of the density matrix is usually unknown. Density matrix of a free particle. In the absence of an external potential the Hamiltonian of a one-dimensional particle with mass and temperature

Let us assume that the particle is confined to an interval [–L/2, L/2]. Then the eigensolutions are given by

Therefore we obtain for the density matrix

In the limit the discrete wave number the summation can be rewritten as an integral

will become continuous and

and thus the density matrix of a free particle becomes

Path integral quantum Monte Carlo. The basic idea of the path integral quantum Monte Carlo method is to express the density matrix of any given system in terms of the free particle density Eq.(4.111). This is justified by the following integral transformation

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Thus in the equation above the density matrix for a system with temperature T will be computed with the help of the density matrix of a system with double the original temperature. But the higher the temperature, the smaller the effect of the potential and thus as better the approximation above. Due to the derivation above we can rewrite the density matrix by

The number f of intermediate steps is called the Trotter number, which should be chosen such that the temperature becomes high compared to the energy spacing. For a particle in an external potential Û the following approximation turned out to be useful

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This approximation would hold exactly for commuting operators For the diagonal element we obtain

and Û.

with

The equations above are derived for a single particle, but could be easily generalized to particles. In this case becomes a dimensional vector and the pre-factor in Eq.(4.114) have to be taken to the power of Quantum statistical averages, especially the partition function2 Z

can be obtained by playing the classical Monte Carlo game. All we have to . f-vector with f the Trotter do is to replace the n-particle vector by the number. Thus the algorithm is: Start with a random position The external potential Û of a particle will be reinterpreted as

with / the Trotter number, and given by Eq.(4.114). The random path is given by displacing the 3f-dimensional vector by and each of the 3-dimensional subvectors < f – 1, by The new configuration is By computing a Metropolis decision is initiated. Let be a random number in [0,1]. If

the trial displacement will be accepted, thus Repeat until the arithmetic mean is converged. For more details about path integral quantum Monte Carlo see, e.g., [18].

Notes l

References

131

2 The partition function is given by

and thus the energy expectation value by

Therefore we can also use the partition function to obtain the energy expectation value.

References [1] Pekeris, C. L., (1958) “Ground state of two-electron atoms”, Phys. Rev. 112, 1649 – 1658 [2] Ericsson, T., and Ruhe, A., (1980) “The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems”, Math. Comput. 35, 1251 – 1268 [3] Saad, Y., (1992). Numerical Methods for Large Eigenvalue Problems: Theory and Algorithms Wiley, New York [4] Bethe, H. A. and Jackiw, R., (1986). Intermediate Quantum Mechanics Addison-Wesley, Reading [5] Lindgren, I. and Morrison, J. (1986). Atomic Many-Body Theory SpringerVerlag, Berlin [6] Kohn, W. and Sham, L. J. (1965) “Self-consistent equation including exchange and correlation effects”, Phys. Rev. 140, A l133 – a1138 [7] Lundqvist, S. and March, N. (1983). Theory of the inhomeogenous electron gas Plenum, New York [8] Perdew, J. and Zunger, A. (1981) “Self-interaction correction to densityfunctional approximations for many-body systems”, Phys. Rev. B 23, 5048 – 5079 [9] Dreizler, R. M. and Gross E. K. U. (1990). Density functional theory Springer Verlag, Berlin [10] James, F., (1980) “Monte Carlo theory and practice”, Rep. Prog. Phys. 43, 1145 – 1189

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[11] Ceperley, D. M. and Bernu, B., (1988) “The calculation of excited state properties with quantum Monte Carlo”, J. Chem. Phys. 89, 6316 – 6328 [12] Jones, M . D., Ortiz, G., and Ceperley, D. M., (1997) “Released-phase quantum Monte Carlo”, Phys. Rev. E 55, 6202 – 6210 [13] Kalos, M. H., Levesque, D., and Verlet, L., (1974) “Helium at zero temperature with hard-sphere and other forces”, Phys. Rev. A 9, 2178 – 2195 [14] Kalos, M. H. (Ed.) (1984). Monte Carlo Methods in Quantum Problems Reidel, Dordrecht [15] Ceperley, D. and Alder, B., (1986) “Quantum Monte Carlo”, Science 231, 555 – 560. [16] Feynman, R. P. and Hibbs, A. R. (1965). Quantum Mechanics and Path Integrals McGraw-Hill, New York [17] Gutzwiller M. C. (1990). Chaos in Classical and Quantum Mechanics Springer Verlag, Berlin [18] Ceperley, D. M. (1995) “Path integrals in the theory of condensed helium”, Rev. Mod. Phys. 67, 279 – 355

Chapter 5 FINITE DIFFERENCES

The equations and operators in physics are functions of continuous variables, real or complex. Computers are always finite machines and hence can only provide rational approximations to these continuous quantities. Instead of representing these values with the maximum accuracy possible, they are usually defined on a much coarser grid and each single value with a smaller resolution, specified by the user in dependence of the problem under consideration and the computing facilities available. In this chapter we will discuss some methods for discretizing differential operators and integrating differential equations. We will not discuss in detail the numerous solvers available for initial value problems of ordinary differential equations, but list them briefly and focus on methods useful for the partial differential Schrödinger equation.

1.

INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

Solvers for initial value problems are divided roughly into single-step and multi-step, explicit and implicit methods. Single-step solvers depend only from one predecessor and -nomen est omen- multi-step solvers from further steps computed “in the past”. Explicit methods depend only from already calculated, implicit methods in addition from the present solution. In the following we will denote the coordinate at time by

and the time step by 133

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and we always assume that the differential equation under consideration is formulated by

Of course we can rewrite any ordinary differential equation

as a system of first-order equations by making the substitutions

1.1

SIMPLE METHODS

THE EULER TECHNIQUE 1.1.1 The Euler methods are based on the definition of the functional derivation

Therefore the explicit Euler method is given by

and the implicit Euler method by

Implicit methods are often used to solve stiff differential equations. A mixture of both is the trapezoidal quadrature

Although these methods seem to work quite well, they are for most systems of differential equations unsatisfactory due to their low-order accuracy. Higher order methods offer usually a much more rapid increase of accuracy with decreasing step size. One class are derived from Taylor series expansions for about

From we obtain

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135

and thus we arrive at

1.1.2 2ND ORDER DIFFERENTIAL EQUATIONS A simple method to integrate second order differential equations

is given by the Verlet algorithm. The Verlet algorithm is explicitly time invariant and often used in molecular dynamics simulations. Starting point is again the Taylor expansion

The Verlet method above is often reformulated in its leap-frog form . To derive the leap-frog algorithm we start with the inter-step velocity

and and thus Note, that this leap-frog form is exactly equivalent to the Verlet algorithm and thus also of 4th order.

2. 2.1

THE RUNGE-KUTTA METHOD DERIVATION

The Runge-Kutta algorithm is one of the most widely used class of methods. To explain the basic idea we will derive the 1st order Runge-Kutta algorithm and list the 4th order Runge-Kutta. The advantage of these single step algorithms are the possibility to optimize easily the step size after each single step. Thus we will in addition discuss the Runge-Kutta-Fehlberg method, which allows optimizing the step size and minimizing the necessary computations.

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is an exact solution of the 1st order differential equation (5.3). This integral can be approximated by the mid-point of the integration interval

From the Taylor expansion

we obtain the 1st order Runge-Kutta algorithm

Runge-Kutta algorithms are explicit single-step algorithms and the order version is exact in order, thus the error is of the order One of the most widely used versions is the 4th order Runge-Kutta algorithm. 4th order Runge-Kutta algorithm. A 4th-order algorithm has a local accuracy of and has been found by experience to give the best balance between accuracy and computational effort. The necessary auxiliary calculations are

to yield the next step

In general the auxiliary variables are given by

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137

with

and the next step by

with

The accuracy in the optimize the integration step

2.2

is of the order

Thus how could we

THE RUNGE-KUTTA-FEHLBERG METHOD

In Runge-Kutta

respectively

with unknown error coefficients

and

order the solution hold

Because

Let be the maximum error allowed. Thus

and hence

The advantage of the method above is the possibility to optimize the integration step size. The disadvantage is that we have to evaluate instead of -times. This problem could be overcome by selecting Runge-Kutta algorithms, which use in both orders the same auxiliary variables. The RungeKutta-Fehlberg algorithm [1] is an example of such a combination for 4th and 5th order.

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The auxiliary variables are

and the successors are

The error

is given by

and thus the new optimized step size by

where we have reduced the allowed error for security by a factor of 0.9. Hence the actual computational path is: 1st select an initial step size and compute If < compute the next step, in case the new step size should be larger the old one, recompute with the actual error and the new step size. Control of typing errors. Because in many publication the coefficients are erroneous due to typing errors1 the Runge-Kutta formula should be always checked. This can be done by the general equations derived in the section above and by testing the algorithm by comparison with a system with a well known exact solution.

Finite Differences

3.

139

PREDICTOR-CORRECTOR METHODS

Multi-step algorithms are algorithms for which the next quadrature step depends on several predecessors. The advantages are their high accuracy and their numerical robustness. The disadvantage is, that it is not as easy as for single-step algorithms to optimize the step size adaptively. Because the next step needs several steps computed in the past, changing the step size necessitates to recompute a new set of predecessor points. Usually this could be done with interpolation routines based on polynomial approximations. In this context Lagrange interpolation polynomials have the advantage of equidistantly spaced nodes, and thus the expansion coefficients of this approximation equals the predecessor values In general a m-step algorithm is defined by

and called explicit otherwise implicit. The most important are the order Adams-Bashforth algorithm: the order Adams-Moulton algorithm: for and thus this is an implicit algorithm. The order Gear algorithm: for Thus the Gear algorithm is as well implicit and useful especially for stiff differential equations. Each of the coefficients are selected such that a order polynomial expansion would be exact. In general the error coefficients of implicit algorithms are smaller than those of explicit algorithms. Implicit algorithms are usually used in combination with explicit algorithms, the so-called predictor-corrector method. Predictor-corrector methods are numerically robust and highly accurate. The essential idea is first to approximate - to predict - by an explicit algorithm the next step. This step is then used in the implicit algorithm - the corrector - instead of the implicit contribution to obtain finally the next step algorithm need predecessors. Thus multi-step algorithm have to be combined with other methods, e.g. a Runge-Kutta algorithm, to obtain the necessary first steps. A typical predictor-corrector combination is the Adam-Bashforth formula 4th order as predictor

and the 4th order Adam-Moulton algorithm as corrector

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The algorithms for ordinary differential equations described in the previous two sections above can be used, e.g., to compute the one-dimensional or the radial solution of scattering states, formulated as initial value problem. We will discuss this aspect more closely in section 4.5 about the Numerov method. They are also useful in context of finite elements, for which we could also obtain systems of ordinary differential equations.

4.

FINITE DIFFERENCES IN SPACE AND TIME

In this section we study discretization techniques by an unidimensional example [3].

4.1

DISCRETIZATION OF THE HAMILTONIAN ACTION

Starting point is the time-dependent one-dimensional Schrödinger equation (1.3). With and we obtain

which will be mapped onto a finite difference equation. The discretized wave function will be indexed in the following way

with the time-index and the space index expansion with respect to the space coordinate gives

A Taylor

where will lead to the discretization point and is the discretization point with step size Thus the Taylor expansion with respect to the space coordinate leads to (we suppress for the moment the time index)

and thus in analogy to the Verlet algorithm

and

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Therefore the Hamiltonian action on the space difference grid is given by

with

4.2

DISCRETIZATION IN TIME

Straight forward discretization of the time variable of the Schrödinger via Taylor expansion results in a non-unitary algorithm. Therefore we will use the Cayley method as described in Chapt. 1.4.2.1. From Eq.(1.58) we obtain with the time step

4.2.1 KICKED SYSTEMS Suppose we want to study a system with a potential pulse (or a potential step) at time T. In this case we have to guarantee that we hit the system exactly at the time and thus T have to be an integer factor of the time step The wave function before and after the pulse (or step) can then be computed in a Cayley-like way. This could be understood easily by a simple example.

Let be the wave function just before the pulse and immediately after the pulse, with The Schrödinger equation

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for a single pulse is given by

and thus the wave packet propagation by

and therefore with

Hence the practical computation, see Fig.(5.1), runs in the following way: For compute by Eq. (5.34). This will give us finally By Eq.(5.37) we obtain then and for the wave function will be again computed by Eq.(5.34).

4.3

THE RECURRENCE-ITERATION

Putting the pieces together and reordering the relevant equation will give us a simple procedure to compute wave packet propagation for a one-dimensional time-independent system. From Eq.(5.34) we obtain the time iteration procedure and from Eq.(5.32) we know how the Hamiltonian act on the finite difference grid. Both together leads to

With the abbreviation

we get

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and by defining the righthand side as

and with the ansatz

and thus

By construction and therefore

Taking the inverse and reordering these two equations gives

Thus for time independent potentials we can skip the upper index

becomes time independent and thus

As boundary condition we assume

for all time steps. Therefore we have to select a sufficiently large space area, such that the wave packet becomes zero at the boundary and we have to restrict the time interval to such a range, that this boundary conditions remain fulfilled (in the numerical sense). From the original ansatz, Eq.(5.41), we get

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Therefore we obtain from the lefthand side boundary condition

and from the righthand side boundary condition

and thus and by definition Thus we get the following computational structure: We know from the previous run and because of course the potential is given and we can compute from the initial wave packet at the beginning. From the wave packet at time step we get and thus we know and can compute all iteratively. Now we have all quantities available to compute the wave packet of the next time step which we will do ‘spacely upside down’. Due to the boundary condition we know that

and all other wave function values on our finite difference space grid can be computed iteratively in inverse order via

which completes the computation for time step and thus we are now ready to compute the next time step The computation above can be generalized to any spatial dimension. But note that only in one spatial dimension explicite equations can be derived. For higher dimensional systems implicit linear algebra equations remain. For conservative systems the relevant matrices are time-independent and thus for fixed time steps some linear algebra tasks like matrix decompositions have to be carried out only once. 4.3.1

EXAMPLE

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As a graphic example we will discuss wave packet propagation through a potential wall. As wall we selected a step-like symmetric potential. Of course our potential could have any shape. The only important point is, that the spacial resolution is sufficiently high. In our example the basic line has the length L = 1 and the total width of the potential is 0.2. Thus each stair has a length of 0.04. The number of spatial steps is 2000, which is fairly sufficient to resolve this potential. The ratio between the squared spatial and time resolution is given by Eq.(5.38), It turned out that is numerically stable. The group velocity of a wave packet is given by

with L/2 the distance the wave packet could travel under the assumption, that its tail vanishes approximately at the spatial border and T the total time. From the equation above we obtain

and thus

Of course these equations are only rough estimates to obtain the correct order of the relevant values and have to be checked numerically. Some results are presented in Fig.(5.2).

4.4

FINITE DIFFERENCES AND BOUND STATES

The Hamiltonian action on the discretized wave function is given by Eq.(5.32). In this section we will derive a simple equation to compute the Hamiltonian eigensolution by a finite difference grid. Bound states are square integrable and thus have to vanish numerically outside a certain space area. Let us discuss for simplification a uni-dimensional system with eigenvectors

Thus we obtain the following finite difference equation for the eigensolutions

This corresponds to a matrix equation

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with

Thus solving this simple tridiagonal matrix will give us the corresponding eigensolutions. Tridiagonal matrices can be efficiently solved using the QRalgorithm (see Chapt. 3.5.2.2). The advantage of finite difference methods is its simplicity. E.g., using standard software like MATLAB allows to derive the relevant programs in only a few hours. The disadvantage of the approach derived above, is its fixed step size. Even if this could be generalized to optimized non-constant step sizes we will not stress that point further. The finite difference method should only be used for oscillator-like potentials and to get a rough first estimate. For a more detailed study or in cases for which this simple approach converges only hardly, other methods, like finite elements, are more valuable. For separable 3-dimensional systems for each of the single differential equations finite difference methods can be used. E.g., for the radial Schrödinger equation we get

By

the radial Laplace-Beltrami operator can be mapped onto For other coordinate systems, e.g. cylindrical coordinates, the Laplace-Beltrami operator remains mixed of first and second order derivations and thus the equation for the discretized wave functions becomes more complicate and the convergence weaker. Thus again, for higher dimensional systems the finite element method becomes more favorable. 4.4.1 EXAMPLES As simple examples we will discuss the Pöschl-Teller potential [4], and the harmonic and anharmonic oscillator. In contrast to the oscillator potentials the Pöschl-Teller potential is of finite range and thus ideal for the finite difference technique, because the wave function becomes exactly zero at the singularities of the potential.

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The Pöschl-Teller potential is given by

with and potential parameters. We will restrict the following discussion to and with (Of course it would be no computational problem to obtain additional results for other values.) The potential (5.57) is shown in Fig.(5.3). For the Pöschl-Teller potential is symmetric around its minimum and possesses for the values above singularities at and Thus the wave function becomes zero at these points and vanishes outside the potential. By some tricky substitutions the corresponding Schrödinger equation can be mapped onto a hypergeometric equation [4] and the exact eigenvalues are given by

In Fig.(5.3) we present the lowest five eigenfunctions and in Tab.(5.l) the eigenvalues in dependence of the number of finite difference steps.

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By computing the potential and the wave function at only 7 steps the ground state energy shows already a remarkable accuracy. It differs only by 3.5% from the correct value. Of course the eigenenergy of the higher exited states are - as expected - less accurate. Using a finite difference grid of 25 steps gives already reasonable eigenvalues for the first five eigenstates. The deviation of the 5th eigenstate from its correct value is approximately 3.5%. For hundred steps the accuracy of the ground state energy is of the order of and for the 5th excited state of about 0.1%. Thus with only 100 finite difference steps the lowest eigenvalues and its eigenfunctions can be sufficiently accurate computed and this will take, even on very small computers, only a few seconds. As a second example we will briefly discuss the harmonic and anharmonic oscillator. In those systems the coordinate space becomes infinitely large and thus can only be restricted due to numerical considerations. Therefore we have now to take care of two parameters to obtain converged results: The size of the selected coordinate space and its coarsening due to the finite step size. In the upper part of Tab.(5.2) the step size in increased but the integration border in coordinate space is kept fixed. By increasing the finite difference step the results become less accurate as expected. In the lower part of Tab.(5.2) the step size is kept fixed, but now the size of the space is shrinked. This could have a much more tremendous affect on the accuracy of the results. Outside the border the wave function is set equal to zero and thus this parameter have to be treated with care. Physically this means that the quantum system is confined by impenetrable walls. In Tab.(5.3) some results for an anharmonic oscillator are presented. Again decreasing step size leads to increasing accuracy. The left and the right border have to be selected such, that the wave function computationally vanishes. For non-symmetric potentials there is no longer any reason to choose the borders symmetric. This is documented by the last line in Tab.(5.3), for which the left

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border became smaller, but the accuracy equals for the first 5 eigenvalues the “symmetric computation”.

5. 5.1

THE NUMEROV METHOD DERIVATION

The Numerov method is a particular simple and efficient method for integrating differential equations of Schrödinger type. It could be used for computing eigenstates, resonances and scattering states. The essential idea is similar the derivation for the finite difference method, but now we include in addition 4th order terms in the Taylor expansion of the wave function. This results in a expansion with a 6th order local error. A Taylor expansion of the wave function, Eq.(5.32), with respect to the space coordinate is given by

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and thus with

we obtain

From the time-independent Schrödinger equation

we get

and thus the Numerov approximation becomes

This approximation can be generalized easily to higher space dimensions and can be used in two different ways: Reordering this equation leads to an eigenvalue problem to compute bound states or resonances, by merging the Numerov approximation with complex coordinate rotation. Interpreting this equation as an recursion equation allows to solve initial value problems, thus, to compute, e.g, scattering states by recursion. Programming this recursion equation is rather simple and straightforward. After obtaining the first two steps, the Numerov scheme is very efficient, as each step requires the computation only at the grid points. To obtain the first two steps it might be necessary to start, e.g, with a Taylor expansion and to use at the beginning some additional intermediate steps. How to start. For systems with definite parity, which is rather the rule than the exception, we know already initial values at the origin. Because

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and normalization is only of importance after finishing the recursion, we can always start with for states with positive parity and with

for states with negative parity. Taylor expansion up to 4th order will give us the next step But note, choosing an erroneous parity will of course also lead to a result, a result which has nothing to do with the physical system under consideration. An illustrative example is shown in Fig.(5.4). On the lefthand side the ground state and the first excited state of the harmonic oscillator computed with the Numerov approximation are plotted. On the righthand side the Numerov scheme is used to obtain the results for the harmonic oscillator Schrödinger equation using the ground state energy and the energy of the first excited state, but with the erroneous initial condition of a negative parity state for the ground state and of a positive parity state for the first excited state. Due the symmetry only the positive half space is plotted. The computations with

References

153

the wrong initial condition leads to diverging results of the corresponding differential equation. (But note, not for any system an erroneous initial condition might lead to such an obvious failure.)

Notes 1 Fehlbergs [2] original publication had an typing error (4494/1025 instead of the correct 4496/1025) and the celebrated Runge-Kutta-Nyström 5th(6th) order algorithm was corrected 4 years after publication. is time-independent. Thus it is only necessary to compute 2 Remember these coefficients once.

References [1] Dormand, J. R. and Prince, P. J., (1989). “Practical Runge-Kutta Processes”, SIAM J.Sci.Stat.Comput. 10, 977 - 989 [2] Fehlberg, E., (1969). “Klassische Runge-Kutta-Formeln fünfter und siebter Ordnung mit Schrittweitekontrolle”, Computing 4, 93-106; (Correction in (1970) Computing 5, 184) [3] Goldberg, A., Schey, H. M. and Schwartz J. L. (1967). “Computergenerated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. Journ. of Phys. 35, 177 – 186 [4] Flügge, S., (1994). Practical Quantum Mechanics Springer-Verlag, Heidelberg

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Chapter 6 DISCRETE VARIABLE METHOD

This chapter gives a review to the discrete variable representation applied to bound state systems, scattering and resonance states and time-dependent problems. We will focus on the combination of the discrete variable method with other numerical methods and discuss some examples such as the alkali metal atoms in external fields, which is of interest with respect to astrophysical questions, Laser induced wave packet propagation and quantum chaos. Further examples are the anharmonic oscillator, an application to periodic discrete variable representations, and the discussion of the Laguerre mesh applied to the radial Schrödinger equation.

1.

BASIC IDEA

Suppose that we wish to solve the Schrödinger equation with the Hamiltonian and the ith eigenvalue respectively eigenfunction, by an expansion of our postulated wave function with respect to an orthonormalized complete Hilbert space basis By setting this ansatz into the Schrödinger equation and taking the inner Hilbert space product we arrive in general at an infinite set of equations

For non-integrable Hamiltonians one is forced, in the complete quantum theoretical treatment, to resort to numerical methods. This means the infinite equations have to be restricted to finite equations. Hence expression (6.1) restricted to a finite basis expansion will become equivalent to a truncated matrix representation, respectively eigenvalue problem, and the variational parameters are then usually determined by diagonalization. By increasing the dimension 155

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of the corresponding matrices this procedure leads monotonically to more accurate eigenvalues and eigenfunctions, provided the inner Hilbert space product (which means the single matrix elements) can be computed exactly. Because this procedure is equivalent to a variational process it is called variation basis representation or spectral representation. In the finite basis representation1 the matrix elements are computed by numerical quadrature rather than by continuous integration. If a Gaussian quadrature rule consisting of a set of quadrature points and weights is used to compute the matrix elements, there exists an isomorphism between the finite basis representation and a discrete representation of the coordinate eigenfunctions based on the quadrature points, the discrete variable representation. This equivalence was first pointed out by Dickinson and Certain [1]. Therefore in brief, the discrete variable method is applicable to a quantum system under the condition, that there exists a basis set expansion and that a Gaussian quadrature rule can be used to compute the matrix elements in this expansion. Light et al. [2] have pioneered the use of the discrete variable method in quantum mechanical problems; first applications in atomic physics in strong external fields can be found in Melezhik [3]. A combination of the discrete variable method in the angular space with a finite element method is described in Schweizer et al. [4], applications to the hydrogen atom in strong magnetic and electric fields under astrophysical conditions are discussed in [5] and to Laser induced wave packet propagation in [6]. A pedagogically nicely written paper is [7]. A discussion of the discrete variable method in quest of quantum scattering problems can be found in [8] and in combination with the Kohn variational principle, e.g., in [9, 10]. A computation of the density of states using discrete variables is presented in [11 ]. They reduced the infinite algebraic equations to a finite one by using the analytical properties of the Toeplitz matrix and obtained very accurate results for one-dimensional potential scattering. Of course this is a very individual selection of examples, therefore see in addition the references in the papers cited above. A theoretical description of the discrete variable representation is outlined in the following section. Section 3 gives a brief review of orthogonal polynomials as it is needed for use in the discrete variable method. A more thorough discussion can be found in some textbooks, e.g. [12, 13]. In section 4 we will discuss some examples. Finally, Sect. 5 is devoted to the subject of the Laguerre meshes.

Discrete Variable Method

2. 2.1

157

THEORY ONE-DIMENSIONAL PROBLEMS

For simplicity let us start with one-dimensional problems. The generalization to Cartesian multi-dimensional problems is straightforward. In many situations the symmetry of the underlying physical systems can be exploited more effectively by using symmetry-adopted non-Cartesian coordinates (see Chapter 2). In those situations the discrete variable method have to be generalized leading to non-unitary transformations or have to be combined with other numerical methods. We will discuss this problems and some ansätze in the next subsection. The following discussion will be restricted to onedimensional problems but could be generalized easily to higher dimensional systems in a Cartesian coordinate description. In the variation basis or spectral representation a wave function is expanded in a truncated orthonormal Hilbert space basis

Due to the orthonormality of the basis with respect to the continuous space coordinate variable, the expansion coefficients are given by

By approximating this equation via discretization on a grid with or grid points 2 we obtain

under the assumption that the basis functions

quadrature

remain orthonormal

in the discretized Hilbert space. Now let us start with some linear algebra gymnastic, following closely the ideas of [2], The discretized expansion coefficients equals the expansion coefficients if we use a sufficient high number of grid points, respectively if our integration remains exact. Therefore we assume, that (at least) in the numerical limit our quadrature (6.4) becomes exact. By expanding our wave

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function (6.2) in the discretized frame we obtain

with

and

Hence is a function which acts on the continuous space variable but is labeled by the selected grid points In analogy to Eq.(6.2) defines a basis in the discrete representation which is orthonormal (see discussion below) due to the inclusion of the quadrature weight in the definition. Therefore it makes sense to interpret (6.7) as a transformation from the variation basis representation to the discrete Hilbert space. By choosing both dimensions equal, we avoid linear algebra problems with left- and right-transformation and obtain

with

From the discrete orthonormality relation (6.5) we get

and due to the Christoffel-Darbaux relation for orthogonal polynomials [2, 13]

This second orthonormality relation can also be derived directly: Due to Eq. (6.8) and, because the trace is invariant under commuting the matrix product, we also have Additionally we obtain

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159

therefore is idempotent. Because idempotent matrices have either eigenvalues equal zero or one and because must have N eigenvalues equal one, hence and T is unitary. T was the transformation from the orthonormal basis function in Hilbert space onto the basis functions and hence this set of basis functions is also orthonormal. Now let us consider the kinetic and potential operator and of the Hamiltonian. In the spectral representation we obtain

where this approximation is the finite basis representation already mentioned above, which would be exact if our quadrature is exact. In the coordinate representation the potential operator is simply multiplicative and hence we can define which holds for any position and ergo also for quadrature grid points. Therefore we can transform this operator from the continuous Hilbert space onto the discretized Hilbert space

where F B R denotes the finite basis representation. Because this transformation is unitary we can construct directly the discrete variable representation (DV R) via:

and since T is unitary the discrete variable representation becomes isomorphic with the finite basis representation. Note, that in the discrete variable representation the potential contribution, or more general the contribution of multiplicative operators, is simply obtained by computing the potential at the grid points and the contribution of derivative operators, like the kinetic energy, by acting with this operator on the basis and taking the results at the

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selected quadrature grid points. In general the expansion coefficients of the wave function are related by

(For some examples see subsection 4.)

2.2

MULTI-DIMENSIONAL PROBLEMS

The multi-dimensional generalization of the discrete variable method in a direct product basis is straightforward. In a direct product basis the quantum numbers of the one-dimensional sub-states are not related to each other. That means the wave function corresponding to a specific degree-of-freedom are labeled by indices which are only associated with this specific degreeof-freedom. Therefore the total transformation T for a n-dimensional system is given by a direct product of the transformation for each single degree-of-freedom E.g., for three Cartesian degrees-of-freedom [9] the spacing of the discrete variable grid in each dimension is equidistant and mass-weighted in case the three masses are different. Hence and and the Hamiltonian becomes

with the kinetic energy matrices given by

A counterexample to direct product states are the well known hydrogen eigenstates with the principal quantum number, the angular momentum and the magnetic quantum number. The approach mentioned above cannot be used for multi-dimensional eigenstates with quantum numbers associated to each other. This could be easily understood, because even the dimension of the variational basis representation will not be equal the dimension in the discrete variable basis representation by simply copying the “direct product” approach. Direct product states are rather the exception than the rule. Nevertheless the multi-dimensional grid can be derived by a direct product of one-dimensional grids. The main task is then to derive a numerical integration scheme on this multidimensional grid that preserves the orthonormality of the variational basis function. For many problems the symmetry of the physical system can be exploited most effectively in a nonCartesian coordinate system (see Chapter 2). Since progress in this subject is helped by concrete examples, let us study, as a prototype, spherical coordinates.

Discrete Variable Method

Spherical coordinates. The kinetic energy operator nates is given by

161

in spherical coordi-

with the angular momentum operator

The spherical harmonics are the eigenfunctions of the angular momentum operator with eigenvalues Thus, the spherical harmonics are the most appropriate spectral basis for the angular momentum operator and the rotational kinetic energy operator will become diagonal in this basis. The spherical harmonics are related to the associated Legendre functions by3

where for

we have made use of

Because l and are not independent from each other, the spherical harmonics are not a direct product state. In numerical computations the basis has to be restricted to a maximum angular momentum l, and to a maximum value in the magnetic quantum number and positive. Therefore in the variational basis representation the dimension is provided runs from to Thus, for maximal magnetic quantum number the dimension of the basis becomes For a direct product state the total dimension would be equal the product of the single dimension in each direction, respectively quantum number. Hence a direct relation between a direct product ansatz, as necessary for the discrete variable representation, and the variational respectively finite basis representation is only possible for If the quantum system possesses azimuthal symmetry4 the magnetic quantum number will be conserved and the can be separated off. Therefore the angular wave function can be expanded in a basis of the associated Legendre functions, with the value of the magnetic quantum number m fixed. The associated Legendre function is a polynomial of degree

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multiplied by a prefactor and orthogonal on the interval [–1, 1], Thus the discrete grid points are given by the zeros of the associated Legendre polynomial and the (see Eq.(6.4)) are the corresponding Gaussian quadrature weights for this interval [–1,1]. The variational or spectral basis is given by

Hence in analogy to Eq.(6.2) we obtain

and, see Eq.(6.5), the discrete orthonormal relation

Thus the Hilbert space basis in the finite basis representation, see Eq.(6.7), is given by

and the transformation between the discrete variable representation and the finite basis representation, see Eq.(6.7), by

For systems such that the magnetic quantum number m is not conserved we have to define a two-dimensional grid in the angular coordinates As mentioned above the dimension of both unidimensional sub-grids has to be equal. Because the is not associated with an orthonormal function there is no similar rule how to select the appropriate grid points. Widely excepted, but nevertheless not a strict rule, is an equidistant ansatz for As already discussed, for conserved magnetic quantum number the depends on Hence this angular grid is and because is no longer a conserved quantity the discrete variable representation would be no longer a convenient representation for Hamiltonians without azimuthal symmetry. This complexity can be avoided by choosing an identical angular grid for all values of Thus we choose for the discrete grid points the zeros of

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the Legendre polynomial with corresponding weights given by the N-point Gauss-Legendre quadrature. Following the ideas of Melezhik [3] we define our (non-normalized) angular basis by

To summerize, the nodes of the corresponding grid with are given by the zeros of the Legendre polynomial and with weights Therefore our Hilbert space basis in the finite basis representation becomes

For practical computations it is more convenient to combine the multi-label by a single label and to re-normalize the ansatz during the actual computation. Note, that because the is not related to orthogonal polynomials this ansatz becomes only orthogonal in the limit and hence the corresponding Hamiltonian matrix will be weakly non-symmetric. We will discuss these questions and related problems and we will derive the DVR representation of the Hamiltonian by a concrete example in subsection 4. An alternative way to discretize the two-dimensional angular space is discussed in [14]. Because in their discretization the number of grid points for the azimuthal angle is 2N – 1 for N rotational states (l = 0 ... N – 1), the grid-dimension equals N(2N – 1) and hence is approximately twice the dimension in the finite basis representation. The advantage of such an ansatz is the higher resolution in the azimuthal angle the disadvantage is obviously the different dimensions which does not allow a simple similarity transformation between the Hamiltonians in both representations. TIME-DEPENDENT SYSTEMS, RESONANCES AND SCATTERING STATES In this subchapter we will discuss briefly time-dependent systems and add a few remarks to resonances and scattering states. The basic idea in using the discrete variable representation to describe wave packet propagation is to map the time-dependent Schrödinger equation onto a system of linear equations and to describe the wave packet and the Hamiltonian in the discrete variable representation. 2.2.1

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In order to describe the time evolution of a wave packet, one have to solve the time dependent Schrödinger equation

A formal solution at time t+dt can be written as

In chapter (1.5) we discussed in detail effective methods to compute the time propagation of a wave packet. To apply the discrete variable approach to those methods is straightforward. To demonstrate the principle way in which the discrete variable method can be used, let us restrict the following discussion to conservative quantum systems. One possibility for bound state systems is the expansion of the initial wave packet with respect to the eigensolution of the Hamiltonian. In this case the time dependence is given by the eigenenergies of the system under consideration and the expansion coefficients of the initial wave packet as described in chapter (1.?). On the contrary this method will, e.g., fail if tunneling becomes important or if the number of eigenstates becomes numerically hardly feasible. Hence in many situations it is necessary or at least numerically more stable to solve directly the time-dependent Schrödinger equation. Methods based on Chebychev expansions, the Cayley or CrankNicholson ansatz, or on direct expansions of the propagator and subspace iterations lead to operator equations. Every action of an operator on a wave packet can be mapped onto the action of the corresponding matrix, Eq.(6.14), on the discretized wave packet represented by a vector. Hence we end up with a linear system of equations. An example is discussed in subsection (4.2.3). In many 3-dimensional applications the wave function is expanded with respect to angular and radial coordinates and the discrete variable method is restricted to the angular part. In this case the 3-dimensional Schrödinger equation is mapped onto a system of uni-dimensional differential equations. Hence the discrete variable part does not differ in its application for bound or continuum solutions, because the boundary condition will only affect the radial component. Hence methods like the complex coordinate rotation will only act on the radial component. Ergo taking into account special boundary conditions, e.g. via Numerov integration, is only necessary for the radial component. An example for the combination of discrete variables, finite elements and complex coordinate rotation will be discussed in chapter 6. The disadvantage of orthogonal polynomials in describing continuum solutions is their finite range. In some applications the Hamiltonian system will be placed in a large box with unpenetrable walls or focus on an evaluation of the wave function in a finite range. In all those methods the first step is to find

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an ansatz, which will be able to describe the asymptotic behavior by a finite range evaluation. After formulating such an ansatz the wave packet have only to be described in finite space and hence it is straightforward to use the discrete variable method. The same philosophy holds for time-dependent approaches to continuum solutions.

3.

ORTHOGONAL POLYNOMIALS AND SPECIAL FUNCTIONS

In the previous sections we investigated the discrete variable method. Basic requisites of this discretization technique are orthogonal polynomials and their nodes. Orthogonal polynomials and special functions are playing a crucial role in computational physics, not only in quantum dynamics. In this section we will report about orthogonal polynomials giving preference to discrete variable applications. Nevertheless the following discussion will not be restricted only to this application and will additionally include some remarks about special functions.

3.1

GENERAL DEFINITIONS

Orthogonal polynomials can be produced by starting with 1, and employing the Gram-Schmidt orthogonalization process. However, although this is quite general, we took a more elegant approach that simultaneously uncovers aspects of interest to physicists. The following definitions, lemmas and theorems can be found in [12, 13, 16, 17]. In this section we will denote by [a, b] an interval of the real axis, by a weight function for [a, b] and by an orthogonal polynomial of degree A function is called a weight function for [a, b] if and only if

exists and remains finite for each

and

The numbers

defined above are called moments of The polynomials of degree are called orthonormal polynomials associated with the weight function and the interval [a, b] if and only if

The inner or scalar product between two function f, defined by

on the interval [a, b] is

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Because is a polynomial of degree of course, has n zeros More important with respect to discrete variable techniques is the following theorem: The zeros of the orthogonal polynomial are 1. real 2. distinct 3. elements of the interval [a,b] One important (and obvious) property of orthogonal polynomials is that any polynomial of degree can be written as a linear combination More precisely

By writing the orthogonal polynomial in the following form

a three-term recurrence relation can be derived:

This recurrence relation leads to a remarkable proof of the reality of the nodes of the orthonormal sequence Let us rewrite (6.35) by the following matrix-equation:

Discrete Variable Method

Now suppose we chose x as a node of equation:

167

we arrive at the matrix

and hence the eigenvalues of the symmetric tridiagonal matrix J are the zeros of J is called the Jacobi matrix of the orthonormal polynomial. Since J is a symmetric matrix, all eigenvalues and thus all roots of the corresponding orthogonal polynomial are real. From the matrix equation above we can also derive the Christoffel-Darboux relation

For completeness we will also mention the Rodrigues formula

with a polynomial in independent of The Rodrigues formula allows to derive directly a sequence of orthogonal polynomials via n-times derivation. It is also possible to generate all orthogonal polynomials of a certain kind from a single two-variable function by repeated differentiation. This function is called generating function of the orthogonal polynomial. Assume that a function fulfills

then the orthogonal polynomial

is (up to normalization) given by

Some examples will be listed in the following subsections. One of the most important and useful applications of orthogonal polynomials occurs in quest of numerical integration techniques. Let be a polynomial of degree Then the formula

with weights and the zeros of the orthogonal polynomial is called Gauss quadrature. The generalized Lagrangian interpolation polynomials

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fulfill clearly and hence the quadrature weight can be computed by

The Gauss quadrature is one of the most successful integration techniques in approximating integrals over general functions. In dependence of the selected orthogonal polynomial, which of course depends also from the integration interval, the Gauss quadrature gets the polynomial name as an additional classifying label, e.g., Gauss-Legendre quadrature. In the following subsections we will tabulate important properties and discuss applications and numerical aspects of some selected polynomials.

3.2

LEGENDRE POLYNOMIALS

The spherical harmonics are the eigenfunctions of the angular momentum operator. These functions have an explicit representation which is based on the Legendre polynomials respectively the associated Legendre functions. Of course applications can not only be found in quantum mechanics but also in electrodynamics, gravitational physics, potential theory or any other theory in which the Laplace operator plays an important role. Thus there is a large body of literature about Legendre functions and its applications. The Legendre polynomial is a polynomial of degree l defined by (Rodrigues formula)

with l zeros in the interval [–1,1]. The associated Legendre functions defined by

are

is a polynomial of degree with nodes in the interval [–1,1] and sometimes called associated Legendre polynomial. With respect to the discrete variable method only those interior zeros of the associated Legendre function are of importance. Associated Legendre functions with respect to negative index m are defined by

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169

Legendre functions are useful in such situations in which solutions of the quantum system can be computed efficiently by an expansion with respect to spherical harmonics or, in spite of this discussion, by the corresponding discrete variable expansion. Note, that the efficiency of the discrete variable method does not necessitate that a spherical expansion of the wave function is efficient. Hence, discrete variable techniques are not related to symmetric or near-symmetric situations of the quantum system under consideration. In dependence of the necessary numerical accuracy we have to choose a certain number of grid points, which are given by the zeros of the (associated) Legendre polynomial. Increasing the number of grid points leads to higher order polynomials. In Fig.(6.1) some examples of Legendre polynomials are shown and one important disadvantage of discrete variable techniques is uncovered. Suppose our Hamiltonian potential as a function of shows strong localized peak structures. In such a situation increasing the numerical accuracy necessitates a higher number of grid points only locally. Doubling the number of grid points in one sub-interval doubles also the number of zeros in other areas of the interval and hence we have to pay an unnecessary high computational price. This is shown in Fig.(6.1) top to bottom. On top the polynomial degree increases from 7 to 9 and hence the number of grid points is increased, but in a smooth way, polynomially distributed. On bottom the polynomial degree is roughly doubled which leads to the numerically claimed local increase of nodes, but also to approximately a doubling of zeros in areas were it might not be necessary. Hence in situations in which the potential as a function of the discrete variable coordinate shows only locally strong fluctuations, discrete variable techniques are numerically not the most efficient technique. In such situations it might be more favorable to resort to finite element methods as described in the next chapter. To compute associate Legendre functions the following recursion relation is very useful:

By starting with

and

because

we get secondly

Some examples are labeled in the following equation

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Legendre polynomials are connected with hypergeometric functions by Murphy’s equation

The associated Legendre differential equation is given by

which reduces for m=0 to the Legendre differential equation. It is straightforward to show by Rodrigues equation, that is a solution of the Legendre differential equation for integer n. Legendre polynomials, respectively the associated Legendre functions, hold the following orthogonality relation

By orthonormalizing the Legendre polynomials we arrive at the recursion relation

from which we can derive directly the Jacobi matrix (6.37) to compute the zeros of the Legendre polynomial

By normalizing the associate Legendre functions a similar equation can be derived for the roots of the associate Legendre polynomials. Eigenvalues of tridiagonal matrices can be computed easily by the QL- or QR-algorithm [18]. Hence this is an alternative to the Muller-method which we will explain in subsection (3.6). The geometric and historical background of the Legendre polynomials is due to the expansion of the gravitational potential the radial distance. Suppose we have a point mass at space point A and we want to derive the gravitational potential at point P. The distances to the origin are and and the enclosed angle between Hence the distance between point A and P is thus 1/R could be

Discrete Variable Method

expanded for in powers of and for call this relative variable h, then the coefficients

in powers of of this expansion

171 Let us

are the Legendre polynomials. Thus this equation can also serve as a generating function for the Legendre polynomials The derivative of the associated Legendre function can be computed by

and the parity relation is given by

More equations and relations for Legendre polynomials and associated Legendre functions and their relation to other special functions can be found, e.g, in [19, 20],

3.3

LAGUERRE POLYNOMIALS

Generalized Laguerre polynomials degree in Its Rodrigues equation is given by

are polynomials of

Generalized Laguerre polynomials play an important role as part of the solution of the radial Schrödinger equation

with

The orthogonality relation for the generalized Laguerre polynomials reads

The following recursion relation is very useful, because it allows a numerical efficient and stable evaluation of the Laguerre polynomial for a given index:

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with Similar to the recursion relation for the Legendre polynomial the equation above can be reformulated to obtain a recursion relation for the normalized generalized Laguerre polynomial, which possesses the same zeros as the standard ones. Hence we obtain again a Jacobi formulation which allows computing the zeros. The following equation shows as an example the Jacobi-recursion for the Laguerre polynomial:

and hence the Jacobi matrix becomes

The generating function for the generalized Laguerre polynomial is given by

and the derivative by

Some relations with the Hermite polynomials will be labeled in the next subsection, more equations and relations can be found in many table and quantum mechanic textbooks. Unfortunately in some publications Laguerre polynomials contain an additional factor and some publications make use of the identity Hence in any case one have to take care about the selected definition.

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174

3.4

NUMERICAL QUANTUM DYNAMICS

HERMITE POLYNOMIALS

Hermite polynomials play an important role in quantum mechanics because the eigenfunctions of the harmonic oscillator

are given by

with the Hermite polynomial of n-th order. The Hermite polynomial hold the orthogonality relation

and is defined by its Rodrigues formula

and the generating function reads

The following recursion formula is of importance for computational approach

and By normalizing the Hermite polynomial

we obtain

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175

and hence the Jacobi matrix, whose eigenvalues are the zeros of the Hermite polynomial becomes

The derivative of the Hermite polynomials hold

which leads together with the above recursion formula to

and Thus the Hermite differential equation is given by

are even and are odd polynomials in for the parity of the Hermite polynomial

Therefore we obtain

and they are connected with the confluent hypergeometric function

by

Because the Laguerre polynomial hold

we obtain the following connection between Laguerre and Hermite polynomials of definite parity:

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BESSEL FUNCTIONS

Bessel functions were introduced in science the first time in 1824 from Bessel in context of astronomical questions. In quantum dynamics Bessel functions are of importance with respect to scattering solutions of those systems in which the potential converges quicker than the centrifugal potential, hence In such situations the radial function can be asymptotically written as a linear combination of the spherical Bessel function (see below). The ordinary Bessel function is a solution of the following second order differential equation

and fulfills the regular solution near the origin

This ordinary Bessel function is connected with the spherical Bessel function by

Note, that unfortunately in the literature there are different definitions for the spherical Bessel function. A common definition is also

and thus index n is given by

The derivative of the Bessel function with integer

and fulfills the following recursion relation

All Bessel functions with integer index

with

can be derived from

by

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177

The Rodrigues formula of the spherical Bessel function is given by

and they both fulfill the same recursion relation (similar to the ordinary Bessel function)

The first two functions are given by

Much more relations about integer and half-integer Bessel functions and some integral definitions can be found in [19]. In many situations in which the analytic solution of the Schrödinger equation can be obtained by Bessel functions it is nevertheless less laborious to obtain the solution directly by optimized numerical routines.

3.6

GEGENBAUER POLYNOMIALS

In the coordinate representation the momentum operator becomes a derivative operator and the position operator is multiplicative. In the momentum representation this is vice versa. The eigenfunctions of the hydrogen atom in the momentum representation are discussed nicely in [21]. The Gegenbauer or ultraspherical polynomials characterize the momentum wave function of the Coulomb system in a similar way as the Laguerre polynomials in the coordinate representation. Both are connected with each other by Fourier transformation. The generating function of the Gegenbauer polynomial is

Hence for the Gegenbauer polynomials equal the Legendre polynomials, see Eq.(6.58), and thus the are generalizations of the Legendre polynomials. The recurrence formula for fixed index is

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with and the derivative holds

The orthogonality relation is given by

and its Rodgrigues equation by

and the

are solutions of the following differential equation

More relations, special values and representations by other hypergeometric functions and other orthogonal polynomials can be found in [20].

3.7

COMPUTATION OF NODES

The computations of nodes means the computations of roots of polynomials. One way is to use the Jacobi representation of the recursion relation, see Eq.(6.37). Of course it is straightforward to generalize this equation to orthogonal polynomials, leading to generalized eigenvalue problems. Numerically this means simply to shift the normalization of the polynomial to the generalized eigenvalue problem. Because the normalization matrix is diagonal this is in both cases simply multiplication by numbers and hence there is no advantage between one or the other computational way. One of the numerically most stable ways to compute the eigenvalues of tridiagonal matrices is the QR-decomposition. Once the Jacobi matrix is derived using the relevant recursion relation, the Jacobi matrix J will be decomposed in two matrices with an orthogonal and an upper triangular matrix, By an iterative process we arrive finally on an diagonal matrix Because

with denoting the transposed matrix and as all matrices are orthogonal by construction, and possess the same eigenvalues and hence the

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179

diagonal matrix consists in the eigenvalues of the original Jacobi matrix J. For general symmetric eigenvalue problems QR-decompositions are not very effective because they necessitate steps to obtain the results. For tridiagonal symmetric matrices this is reduced to steps, making the algorithm numerically highly efficient. QR- and QL-decompositions are equivalent, the only difference is that in QL-decompositions the matrix will be decomposed into an orthogonal and lower triangular matrix instead of an upper triangular matrix. QR and QL algorithms can be found in most standard linear algebra libraries, e.g., in the NAG-, Lapack- or the older Eispack-library and additional in many textbooks about numerical methods. Frequently found in the literature is an iterative computation of the characteristic polynomial of the tridiagonal matrix. If denotes the determinant of the n-dimensional Jacobi eigenvalue problem the n-dimensional identity matrix, then the determinant can be computed by

For orthogonal polynomials this approach means simply to recover the original polynomial root problem and is of no computational help. In general the computation of large determinants should be numerically avoided, because the roundoff errors in building differences between large numbers could lead to an erroneous result and hence the algorithm mentioned above could become numerically unstable. In many situations the simplest way is to compute directly the zeros of the orthogonal polynomial or special function under consideration. Perhaps the most celebrated of all one-dimensional routines is Newton’s method. The basic idea is to extrapolate the local derivative to find the next estimate of the correct zero by the following iterative process

The advantage of orthogonal polynomials is that we know how many zeros exist and in which real interval. Nevertheless for polynomials of higher order we might fail. One of the problems is that, due to the derivative, the next estimated point will be the tangent crossing with the x-axis and this could lay outside the definition interval. To overcome that problem it might be necessary to start with a finer and finer coarse grid of starting points in the theoretically allowed interval. A second cause of the fault is to run in a non-convergent cycle due to the same tangent of two following iterative points. Hence it is useful to use more advanced techniques like the Müller method. The Müller method generalizes the secant method in which the next estimate to the zero is taken where the approximating line, a secant through two functional values, crosses the axis. The first two guesses are points for which the polynomial value lies

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on the opposite side of the axis. Müller’s method uses a quadratic interpolation among three points, instead of the linear approximation of the secant method. Given three previous guesses and to the zero of the polynomial the next approximation is computed by the following iteration procedure

In principle this method is not only restricted to orthogonal polynomials but to general functions and also complex roots can be computed.

3.8

MISCELLANEOUS

Different basis representations lead to different structures of the corresponding operator matrix. One of the simplest computational form, besides diagonal representations, are tridiagonal representations. The radial Coulomb Hamiltonian can be transformed directly into such a tridiagonal form. The orthogonal polynomials associated with the corresponding basis are the Pollaczek polynomials [22]. The Schrödinger equation of the Coulomb problem has an exact solution in both spherical and parabolic coordinates (see Chapter 2). The wave functions in both coordinate systems are connected by Clebsch-Gordan coefficients. This coefficient can be expressed in terras of the Hahn polynomials. For details see [13, 16]. Hypergeometric series or functions play an important role in the theory of partial differential equations. They are a generalization of the geometric series defined by

with

the Pochhammer symbol. The derivative of the hypergeometric function is given by

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181

and because of

Hypergeometric functions are related to many special functions and orthogonal polynomials. For more details see, e.g., [19, 20], To obtain starting values for iterative computations the Gauss theorem

is helpful. The hypergeometric differential equation is given by

Well known examples for quantum systems which lead exactly to this hypergeometric differential equation are molecules approximated by a symmetric top, see, e.g., problem 46 in [23]. By substituting and we arrive at a new differential equation

the confluent hypergeometric equation, and its solution is given by the confluent hypergeometric function

Important applications for the confluent hypergeometric function are scattering problems in atomic systems and, beyond quantum dynamics, boundary problems in potential theory. Hypergeometric functions are generalized by

E.g., is connected with the already mentioned Hahn polynomial. More important is the following relation with the Jacobi polynomial

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There are several equivalent definitions, but unfortunately also some different ones in the mathematical literature, e.g, Hence in any case the exact definition has to be checked before using different tables. The Jacobi polynomial follows the three-term recursion relation

and its generating function

with From this equation it is straightforward to compute the lowest two contributions to the recursion relation above. The Jacobi polynomials are the only integral rational solution of the hypergeometric differential equation. Many orthogonal polynomials can be treated as a Jacobi polynomial with special index One of the most important relations are

and the Gegenbauer polynomial

4.

EXAMPLES

This chapter is devoted to some applications of the discrete variable method. The harmonic oscillator and the hydrogen atom are particularly important physical systems. Actually, a large number of systems are, at least approximately, governed by the corresponding Schrödinger equations and an innumerable number of physical systems are back-boned by these two. The eigenfunctions of the harmonic oscillator (Kepler system) are related to Hermite (Legendre) polynomials. Therefore let us start with the Hermite polynomial followed by an example for Legendre polynomials.

4.1

HERMITE POLYNOMIALS AND THE ANHARMONIC OSCILLATOR

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183

As already mentioned above results obtained in the study of oscillator systems are applicable to numerous cases in physics, e.g., vibration of the nuclei in diamagnetic molecules, torsional oscillations in molecules, motion of a muon inside a heavy nucleus, giant resonances in atomic nuclei, to name only a few. As a didactic example for the discrete variable technique we will discuss the anharmonic oscillator with a general forth order potential. The Hamiltonian of the anharmonic oscillator reads

By using the ladder operators

we obtain

and are the prefactors to respectively order perturbation theory we get for the energy

for

In first

and because of the energy does not undergo an alteration due to the cubic potential in first order. For vanishing forth order term the potential curve will undergo a maximum so that the bound states die out at higher energy. Strictly speaking even the low lying states become quasi bound by coupling to the continuum via tunelling. For the discrete variable computation we use the following ansatz for our wave function:

and

the roots of the Hermite polynomial Because the functions are harmonic oscillator eigenstates the matrix elements can

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NUMERICAL QUANTUM DYNAMICS

be computed easily. By this ansatz it is straightforward and a standard textbook example that satisfies the Hermite differential equation. Hence for the harmonic oscillator we obtain

and because

is multiplicative (see also Eq.(6.12))

Thus we have to solve the following eigenvalue problem

Of course it is straightforward to generalize this result for to arbitrary potentials by For even potentials eigenfunctions of even and odd parity do not mix and the computations can be optimized by projecting the ansatz above on the even, respectively odd parity subspace. Hence the dimension of the eigenvalue problem will be reduced from N + 1 to (N + l)/2 for N odd and for N even to (N + 2)/2 for the positive parity and to N/2 for negative parity states. As a concrete example we will first discuss the quartic anharmonic oscillator. For the positive/negative parity eigenstates we have to solve the following eigenvalue equation for the positive parity states

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185

respectively

for the negative eigenstates, and

projected to the positive/negative sub-

space. In Fig.(6.2) we compare some results for the 2nd exited state obtained by 1st order perturbation theory, by DVR with a 3- and a 6-dimensional positive parity matrix and by diagonalization of a positive parity Hamiltonian matrix The results computed by the Hamiltonian matrix and by the 6-dimensional DVR are in excellent agreement. As expected the perturbational results are much worse. In Fig.(6.3) some DVR-results computed with a 11-dimensional matrix are plotted. For comparison the analytic harmonic oscillator results are surprinted. On the lefthand side some results for the cubic anharmonic oscillator are plotted. With increasing anharmonicity constant the potential curve undergoes its extrema at lower and lower energy. On the bottom, only the lowest two eigenstates ‘survive’, all others are in the continuum. This is an example where one have to exercise caution. By construction the DVR-method will produce real eigenvalues, even if only reso-

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nances survive. Nevertheless, resonances can also be computed by combining the discrete variable method with complex coordinate rotation, or by taking the

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187

correct boundary condition into account. In the complex coordinate method the momentum operator is mapped onto and the coordinate operator onto Therefore if and are the relevant matrices we get

E.g. the first excited state for (lefthand side bottom in Fig.(6.3) will have an complex eigenvalue Hence the width of this eigenstate is still sufficiently small. We will discuss some results in context of the finite element technique in the next chapter. On the righthand side of Fig.(6.3) we present some results for In this case the contribution will dominate the potential for sufficiently large distances from the origin and hence all eigenstates remain real. Of course this would not be true for negative which we will discuss as an example in chapter 6.

4.2

LEGENDRE POLYNOMIALS AND COULOMB SYSTEMS

In this section we will apply the discrete variable method to Legendre polynomials and more general to spherical harmonics. Possible applications are beside the top any one-particle Hamiltonian. As an example we will discuss Alkali atoms in external fields. This example has the advantage that it is not only of principle interest but uncovers in addition many computational aspects related to Legendre polynomials. The kinetic energy of a single electron in a magnetic field is given by with the vector field. In the symmetric gauge this leads to three contributions: The field-free kinetic energy the paramagnetic contribution which leads to the well known Zeeman effect and the diamagnetic contribution which becomes important for extremely strong magnetic fields or for high excited states. Hence the Hamiltonian for an one-electron atom with infinite nuclear mass becomes in the symmetric gauge and appropriate units and spherical coordinates

where the magnetic field axis points into the z-direction For the hydrogen atom in external magnetic and electric fields the potential becomes

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with the magnetic induction the electric field strength measured in units of and representing the angle between the two external fields. The energy is measured in Hartrees (27.21 eV) and lengths in units of the Bohr radius Effective one-electron systems like Alkali atoms and Alkali-like ions can be treated by a suitable phenomenological potential, which mimics the multi-electron core. The basic idea of model potentials is to represent the influence between the nonhydrogenic multi-electron core and the valence electron by a semi-empirical extension to the Coulomb term, which results in an analytical potential function. The influence of the non-hydrogenic core on the outer electron is represented by an exponential extension to the Coulomb term [24]:

Z is the nuclear charge, the ionization stage. (For neutral atoms the ionization stage is for single ionized atoms and for multi-ionized atoms The coefficients are optimized numerically so as to reproduce the experimental field-free energy levels and hence quantum defects of the Alkali atom or Alkali-like ion. In Tab.(6.1) we show some parameters

for Alkali-like ions. This model potential has the advantage, that it has no additional higher order singularity than the Coulomb singularity. In many computations, singularities up to order –2 are lifted due to the integral density in spherical coordinates. The form of this model potential is based on the work of [24], more data can be found in [25].

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189

4.2.1 TIME-INDEPENDENT PROBLEMS Let us now start to discuss the discrete variable method applied to the Hamiltonian (6.128) above. Of course we could add in addition further potential terms, e.g., a van-der-Waals interaction. As already mentioned, the Legendre polynomials are related to the eigenfunctions of the angular momentum operator. Therefore a simple computational ansatz is given by using the complete set of orthonormal eigenfunctions of the angular momentum operator

where

and

As discussed in chapter (2.2), for conserved magnetic quantum number can be separated off, and the N nodal points in are given by the roots of the Legendre polynomial In general we design a by choosing N nodal points with respect to by the zeros of and, for the variable N equidistant nodal points in the interval The value of the first eigenfunctions (6.131) at these nodal points defines a square matrix with elements For N odd becomes a Chebycheff system and hence the existence of the inverse matrix is guaranteed [26]. To simplify the computation we expand the wave function in terms of and the radial functions

This expansion has the advantage, that all those parts in the Hamiltonian which are independent of angular derivative operators become diagonal:

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For the “angular-terms” in the Hamiltonian we get

with Separating the radial function in its real and imaginary part leads to coupled uni-dimensional differential equations:

with

This system of differential equations could be written more clearly in a matrixvector notation

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191

and hence finally

In general the coupling matrix K and thus the differential equation system (6.138) will not be symmetric. As we will show this problem can only be partially solved. But let us first discuss the two-dimensional case. Cylindrical symmetry. For systems with cylindrical symmetry the quantum number is conserved and hence the degree-of-freedom can be separated off. In our example this is the case for either vanishing electric fields or parallel electric and magnetic fields. Thus the angular momentum operator can be reduced to

Therefore we have only N grid points instead of grid points In addition our radial wave function becomes real and hence the system is reduced to a system of N coupled uni-dimensional differential equations in

with

Symmetrization. In order to keep the Hamiltonian matrix symmetric, we have to normalize the three-dimensional wave function in the space of the discrete variables

By defining

and in analogy to Eq.(4.2.1)

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and renormalizing the differential equation (6.138) we arrive after some simple algebra gymnastic at

with

and

If we would be able to design grid points and integration weights in such a way that the corresponding Gaussian quadrature would become exact, the rows of the matrix would built an orthogonal system

and hence our normalization matrix (6.144) would become diagonal In case of the Legendre polynomials, respectively associated Legendre function, the Gaussian integration is exact. Hence for cylindrical symmetry, in which the discretization reduces to the Eq.(6.149) becomes exact. Thus the normalization matrix is diagonal and the differential equation system (6.146) becomes symmetric. For the full system the has also to be discretized. is related to trigonometric functions and hence Eq.(6.149) holds only approximately, respectively in the limit Therefore the differential equation system (6.146) is only approximately symmetric. In summary, by using the discrete variable method for spherical harmonics the 3-dimensional Schrödinger equation is mapped onto a system of ordinary differential equations in the radial coordinate. Hence finally we have to solve this radial equation by a suitable method, e.g., by using the finite element method as described in chapter (6).

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4.2.2 SPECTROSCOPIC QUANTITIES So far we have considered only the computation of wave functions and energies. A complete discussion of spectral properties has also to take into account the probability, that an atom or molecule undergoes a transition from one state to another. In this context the dipole strength

becomes important. The dipole strength is by definition the transition matrix element between two states and Related to the dipole strength are the oscillator strength

and the transition probability

with the Bohr radius and the reduced mass. Due to the spherical harmonic in the transition matrix element, only those transitions are allowed for which the magnetic quantum number remains equal (linear polarization) or changes by ±1 (circular polarization). (For more details see, e.g., [27].) Hence from our ansatz (6.133) we get for the dipole strength

Both parts and of the integral can be computed independently. The parameter q distinguishes between the different possible polarizations. For q = 0 we have linear and for q = ±1 right and left circular polarization. To compute the angular part we rewrite the ansatz (6.131) in terms of the spherical harmonics

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Ergo, computing the angle integral will be reduced to integrate products of three spherical harmonics and hence well-known expressions from the theory of angular momentum coupling can be used. Here we will make use of the Regge symbols [28]. Of course it is straightforward to use or Clebsch-Gordan coefficients instead of Regge symbols.

This representation has the advantage that the symmetries of the angular momentum coupling are best uncovered. Only if all elements are non-negative integers and if all columns and rows are equal the sum of the three coupling angular momenta, the Regge-Symbol [ ] does not vanish. Of course in our case this leads immediately to the dipole selection rules Therefore we get the following results: For

for

and for

Hence with this expressions we can compute directly

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4.2.3 TIME-DEPENDENT PROBLEMS In this chapter we will only add a few remarks to time-dependent problems. E.g., in order to describe the time evolution of a Laser-induced wave packet we have to solve the time dependent Schrödinger equation

where H is the non-relativistic single particle Hamiltonian of an alkali atom or ion in external fields. (Of course the following approach is not restricted to that example. But we will continue this discussion in chapter 6, where we will use finite elements to solve the linear system of uni-dimensional differential equation, derived below.) For the spatial integration we use the combined discrete variable and finite element approach above-mentioned. A solution for the time development of the discretized wave functions is given by:

which can be approximated, e.g., by the Cayley or Crank-Nicholson method

Inserting (6.161) into the time-evolution (6.160) leads to an implicit system of algebraic equations

which must be solved for each time step operator equations of the form

Hence in general we are guided to

with f, polynomial functions. (This equation includes also Chebycheff approximations.) Therefore applying the discrete variable method is straightforward and acts pointwise in time. Thus, instead of the eigenvalue problem for time-independent questions, we get a linear system of equations, which have to be solved in each time step. For example from Eq.(6.162) we will get in analogy of the eigenvalue equation (6.146)

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For explicitly time-independent Hamiltonians the computations are simplified because the Hamiltonian matrix remains invariant under time and hence, e.g., LU-decompositions of the Hamiltonian matrix have to be done only once. In chapter 6 we will continue with this discussion and we will present some results of these time-dependent and time-independent examples by solving the radial part via finite elements.

4.3

PERIODIC AND FIXED-NODE DISCRETE VARIABLES

The following discussion is based on [7]. We start with defining a basis set, not necessarily orthogonal polynomials. The basic idea of this approach is that the defined discrete variable basis will have all nodes spaced evenly in a fixed selected coordinate interval in contrast to the “standard” orthogonal polynomial approach. Let our original Hilbert space basis and coordinate functions such that For a complete Hilbert space basis, we can expand our unknown coordinate function with respect to this basis

and get by a Gaussian integration

Hence, by construction, we obtain

or

This equation is the central result for this approach. Therefore there are two steps left: The selection of the Hilbert space basis and a convenient quadrature rule. Of course there is no general way how to select both, but this freedom allows to incorporate special features of the system under consideration. Because the Hilbert space basis can be defined exactly, the DVR basis is in addition independent from any numerical procedure, Let us now turn to two examples [7] to clarify the approach presented above.

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Fixed-node boundary conditions. As an example for fixed-node boundary conditions we select a quantum system which can be described accurately by a Hilbert space basis built by particle-in-box eigenfunctions

To fit any other boundaries this interval can be transformed by shift and rescaling, With this basis we get from Eq.(6.168)

To find an appropriate quadrature rule, which will give us the relevant nodes and weights let us first discuss the Gauss-Chebyshev quadrature of the second kind [19]. The Gauss-Chebyshev quadrature of the second kind is defined by

where the nodes are given by the roots of the Chebyshev polynomials of the 2nd kind, and the weights by Guided by the fact that sin is the lowest order function not included in the original basis, Eq.(6.169), and has evenly spaced zeros

we will reformulate the quadrature given above to derive the correct weights With we get for an arbitrary integrable function f

with

Discrete Variable Method

Hence by using this quadrature rule we get equally spaced nodes weights become Therefore finally we arrive at

199

and the

An example of this fixed-node discrete variable basis for N = 3 is presented in Fig.(6.4). Periodic discrete variables. basis

In this example we start with the Hilbert space

with and discrete variable basis has to fulfill that becomes

and N odd integer. Because our it is straightforward to show

By using the Gauss-Chebyshev quadrature of the first kind

we obtain with

for an arbitrary integrable function f

and

An example of this fixed-node discrete variable basis for N = 3 is presented in Fig.(6,4).

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5.

THE LAGUERRE MESH

Similar to the discrete variable computation general mesh calculations make use of Gaussian integration techniques and hence avoid the necessity of calculating complicated matrix elements. The basic idea is to use generalized Lagrange interpolation polynomials based on orthogonal functions (see Eq.(6.43)). In the following we will first describe the general idea but restrict the application to Laguerre meshes and unidimensional systems. Using meshes based on other orthogonal functions or polynomials as well as the generalization to multidimensional problems is straightforward. Nevertheless in most cases a combination of discrete variable and finite element systems, as described in the next chapter, seems to be more accurate and efficient. Applications of mesh methods can be found in nuclear [29], atomic [30, 31] and molecular physics. In the following we will discuss mesh calculations for a one-dimensional Schrödinger equation with a local potential and eigensolutions, which approximately vanish outside an interval [a, b]

The approximation of the wave function is based on its values at the mesh points

thus

For generalized Lagrange interpolation polynomials, based on orthogonal functions Eq.(6.43), the Gaussian quadrature is given by

with the roots of Hence let us consider the corresponding family of orthogonal polynomials with a weight function and norm A set of functions normalized on the interval [a, b] is given by

and the mesh points by the zeros of Note, in contrast to the discrete variable technique, these computations are only based on the highest normalized orthogonal polynomial of the selected set A discussion of generalized meshes, including Hermite meshes can be found in

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201

[32]. The following discussion will be restricted to interpolation functions based on Laguerre polynomials. For Laguerre meshes the generalized Lagrange interpolation polynomials are based on

Hence the mesh points will be given by the zeros of and therefore will be different for each (angular momentum) value l. Because the coordinate is restricted to positive definite values, we expect the corresponding mesh to be useful for radial or polar coordinates and hence most applications are given by the radial Schrödinger equation. Thus our kinetic energy operator for a given orbital angular momentum l is given by

and its matrix elements by

with the corresponding Christoffel symbols

Making use of the Gauss-Laguerre quadrature [32] leads to

with

for the kinetic energy, and for the potential matrix elements we obtain

One of the advantages of the Laguerre mesh method is, that the centrifugal potential in the kinetic energy is taken correctly into account. By introducing a coordinate scaling to optimize the convergence, the discretized Schrödinger equation becomes

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Hence solving eigenvalue equation (6.192) will give us the eigenenergies and as eigenvectors the corresponding wave functions at the mesh points. In some applications, e.g. in cases in which the potential is dominated by a modified Laguerre mesh [31] with mesh points defined by

is more appropriate. In this case the interpolation polynomials are given by

with

and

It is of interest to compare the approximative behavior for large of the generalized Lagrange interpolation functions. For the Laguerre mesh the approximative behavior for large is given by due to Eq.(6.185) and for the “squared mesh” by due to Eq.(6.195). The approximative behavior of the radial hydrogen eigenfunction with principal quantum number n is Thus for a scaling factor both, the hydrogen eigenfunction and the interpolation function of the Laguerre mesh, show exactly the same approximative behavior. In contrast to the hydrogen atom the approximative behavior of harmonic oscillator eigenfunctions for large is given by Hence the Laguerre mesh is much better adopted to Coulombdominated potentials, whereas the modified Laguerre mesh, Eq.(6.193), is expected to be more appropriate for parable-dominated potentials. An example for the hydrogen atom in extremely strong magnetic fields will be presented below. Let us consider the kinetic energy for a two-dimensional problem in polar coordinates

hence [31]

for the modified Laguerre mesh.

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Example. The hydrogen atom in extremely strong magnetic fields is an example where both, the modified Laguerre mesh and the ordinary Laguerre mesh, can be used. The Hamiltonian of the hydrogen atom in strong magnetic fields in spherical coordinates is given by Eqs.(6.128) and (6.129). In cylindrical coordinates we obtain in the non-relativistic limit

with the magnetic field strength measured in units of Tesla, and neglecting the conserved Zeeman contribution the electron spin. The eigenfunction of the free electron in an external magnetic field is given by the Landau function [33]

Thus the solution for the hydrogen atom in strong magnetic fields can be expanded with respect to the Landau function for dominant magnetic fields. This is the case for neutron star magneticfields or for high excited Rydberg states with the principal quantum number). In Tab.(6.2) we compare results for the ground state of the hydrogen atom for obtained by a product ansatz for the wave function. In the we used a finite element ansatz and in the a Laguerre respectively a modified Laguerre mesh. The modified Laguerre mesh is in coincidence with the Landau function (6.199).

In semiparabolic coordinates the Schrödinger equation for the hydrogen atom in strong magnetic fields becomes

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with

and this equation is in exact agreement with the kinetic energy operator (6.197). Thus the generalized Laguerre mesh seems more appropriate. Ergo the optimized mesh depends in addition from the selected coordinates. The wave function converges for large distances in the semiparabolic coordinates like but because this is equivalent to the Coulomb-like behavior, mentioned above, In the next chapter we will discuss another discretization technique, the finite element method. In the discrete variable method or the mesh technique the basis is defined globally, which means in the whole space, but considered locally at the nodes. Finite elements go one step further. The space will be discretized by elements, small areas in space, and on each of this elements a local coordinate system possessing interpolation polynomials. Hence both, the coordinate space and the approximation of the wave function, will be considered locally. Again this ansatz will lead to a (symmetric) eigenvalue problem, whose solution will give the eigenenergy and corresponding wave function of the quantum system under consideration.

Notes 1 This somewhat unfortunate naming is due to the finite number of quadrature points in coordinate space. Loosely speaking, in the finite basis representation the coordinate space is represented by the finite number of quadrature points, in contrast to the variation basis representation in which the coordinate space remains continuous. 2 To distinguish between continuous and discretized variables, functions and so on we will use tilded letters and symbols for the discretized ones. 3 Unfortunately different authors use different phase conventions in defining the 4 Systems with cylindrical or spherical symmetry have azimuthal symmetry, which means symmetric under rotation around the azimuthal (z) axis.

References [1] Dickinson, A. S., and Certain, P. R. (1968). “Calculation of Matrix Elements for One-Dimensional Problems,” J, Chem. Phys. 49,4209–4211 [2] Light, J. C, Hamilton, I. P., and Lill, J. V. (1985). Generalized discrete variable approximation in quantum mechanics,” J. Chem. Phys. 82, 1400– 1409

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[3] Melezhik, V. S., (1993). “Three-dimensional hydrogen atom in crossed magnetic and electric fields,” Phys. Rev. A48,4528–4537 [4] Schweizer, W. and P. (1997). “The discrete variable method for non-integrable quantum systems,” Comp. in Phys. 11, 641–646 [5]

P. and Schweizer, W. (1996). “Hydrogen atom in very strong magnetic and electric fields,” Phys. Rev. A53, 2135–2139

[6]

P., Schweizer, W. and Uzer, T. (1997). “Numerical simulation of electronic wave-packet evolution,” Phys. Rev. A56, 3626–3629

[7] Muckerman, J. T. (1990). “Some useful discrete variable representations for problems in time-dependent and time-independent quantum dynamics,” Chem. Phys. Lett. 173, 200–205 [8] Lill, J. V., Parker, G. A., and Light, J. C. (1982). “Discrete variable representations and sudden models in quantum scattering theory,” Chem. Phys. Lett. 89,483–489 [9] Colbert, D. T., and Miller, W. H. (1992). “A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method,” J. Chem. Phys. 96,1982–1991 [10] Groenenboom, G. C., and Colbert, D. T. (1993). “Combining the discrete variable representation with the S-matrix Kohn method for quantum reactive scattering,” J. Chem Phys. 99, 9681–9696 [11] Eisenberg, E., Baram, A. and Baer, M, (1995). “Calculation of the density of states using discrete variable representation and Toepliitz matrices,” J. Phys. A28, L433-438. Eisenberg, E., Ron, S. and Baer, M. (1994). “Toeplitz matrices within DVR formulation: Application to quantum scattering problems,” J. Chem. Phys. 101, 3802–3805 [12] Szegö, G., (1975). Orthogonal Polynomials. Ameri. Math. Soc., Providence, Rhode Island [13] Nikiforov, A. F., Suslov, S. K., and Uvarov, V. B. (1991). Classcial Ortgogonal Polynomials of a Discrete Variable. Springer- Verlag, Berlin [14] Corey, G. C., Lemoine, D. (1992). “Pseudospectral method for solving the time-dependent Schrödinger equation in spherical coordiantes,” J. Chem, Phys. 97, 4115–4126 [15] Kosloff, R., (194). “Propagation methods for quantum molecular dynamics,” Annu. Rev. Phys. Chem. 45, 145-178

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[16] Nikiforov, A. F., and Uvarov, V. B., (1988). Special functions of mathematical physics Birkhäuser, Basel [17] Tricomi, F. (1955). Vorlesungen über Orthogonalreihen. Springer, Heidelberg [18] Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992). Numerical Recipies. Cambridge University Press [19] Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions. Dover Publications, Inc., New York [20] Gradstein, I. and Ryshik, I. (1981). Tafeln - Tables. Verlag Harry Deutsch, Frankfurt [21] Hey J. (1993). “On the momentum representation of hydroenic wave functions: Some properties and an application”, Am. J. Phys. 61, 28 – 35 [22] Broad, J. T. (1985). “Calculation of two-photon processes in hydrogen with an basis” Phys. Rev. A31, 1494–1514 [23] Flügge, S. (1994). Practical Quantum Mechanics. Springer-Verlag, Berlin [24] Hannsen, J., McCarrol, R. and Valiron, P. (1979). “Model potential calculations of the Na-He system”, J. Phys. B12, 899–908 [25] Schweizer, W., Faßbinder, P. and González-Férez, R. (1999). “Model potentials for Alkali metal atoms and Li-like Ions,” ADNDT 72, 33–55 [26] Berezin, I. S. and Zhidkov, N. P. (1966). Numerical Methods, Nauka, Moscow (in Russian), (German translation in VEB Leipzig) [27] Bethe, H. A. and Salpeter, E. E. (1957). Quantum Mechanics of One- and Two-Electron Systems. Springer-Verlag, Berlin [28] Regge, T. (1958). “Symmetry properties of Clebsch-Gordan’s coefficients”, Nuovo Cim. 10, 544–545 [29] Baye, D. (1997). “Lagrange-mesh calculations of halo nuclei, Nucl. Phys. A 627, 305–323 [30] Godefroid, M., Liévin, J. and Heenen, P.-H. (1989). “Laguerre meshes in atomic structure calculations,” J. Phys. B 22, 3119–3136 [31] Baye, D. and Vincke, M. (1991). “Magnetized hydrogen atom on a Laguerre mesh,” J. Phys. B 24, 3551–3564 [32] Baye, D. and Heenen, P.-H (1986).“Generalized meshes for quantum mechanical problems,” J. Phys. A 19, 2041–2059

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[33] Canuto, V. and Ventura, J. (1977). Quantizing magnetic fields in astrophysics. Gordon and Breach Sci. Pub., New York

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Chapter 7 FINITE ELEMENTS

Partial differential equations are used as mathematical models for describing phenomena in many branches of physics and engineering, e.g, heat transfer in solids, electrostatics of conductive media, plane stress and strain in structural mechanics and so forth. Obviously, for many physical problems the geometry for which a solution of the partial differential equation has to be found is the main source of the difficulty. Take, e.g., a simple wrench. The complex shape of this tool is too complicated to compute directly, e.g., the stress on the top. Thus approximation techniques had to be derived to find solutions of partial differential equations on complicated geometries. These difficulties gave birth to the finite element method in order to solve differential equations arising from engineering problems in the 1950s. The essential idea of the finite element technique is to formulate the partial differential problem as a boundary value problem and then to divide the object under consideration into small pieces on which the partial differential equation will be solved using interpolation polynomials. Thus, even the finite element technique was not developed originally to calculate quantum eigensolutions, it can be applied to partial differential equations of Schrödinger-type and it turned out that it provides a convenient, flexible and accurate procedure for the calculation of energy eigenvalues, for deriving scattering solutions or for studying wave packet propagation. One of the first studies for quantum systems is [1], in which the finite element method was used to obtain continuum states. The computation of one dimensional bound states with finite elements is didactically well explained in [2]. In the following chapter we will discuss the basic mathematical background of the finite element technique followed by applications of one- and two-dimensional finite elements to quantum systems. 209

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1.

INTRODUCTION

The finite element method can be applied to any partial differential equation formulated as boundary value problem. The questions that define a boundary value problem are: What equations are satisfied in the interior of the region of interest and what equations are satisfied by the points on the boundary of that region of interest? In general there are three possibilities. Let be a solution of the partial differential equation. The Dirichlet boundary condition specifies the values of the boundary points as a function, thus

and the generalized von Neumann boundary condition specifies the values of the normal gradients on the boundary

with the outward unit normal, and complex valued functions defined on the boundary. Thus the boundary condition could be either of von Neumann type, or of Dirichlet type or a mixture of both. In the following we are interested in solving the time-independent Schrödinger equation. Bound states are square integrable, scattering and resonance states can be mapped onto square integrable functions by complex coordinate rotation. This means that the solutions we are interested in will vanish approximately outside a suitable selected area and thus the following discussion will be restricted to boundary value problems of Dirichlet type. For a thorough and general discussion of finite elements see, e.g., the excellent monography of Akin [3]. Let be a differential operator and

and with the boundary of the bounded domain and a complex function. For continuous problems the boundary value problem can be formulated by an equivalent variational problem

with the functional

This is also called the weak formulation or the Ritz variational principle.

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Example. example:

211

For a better understanding of this equation let us discuss a simple

Multiplying Eq.(7.3) by a test function we obtain

and integrating over the domain

and by partial integration and using the identity

Thus the variational task is to find a solution u such that Eq.(7.4) holds, where for this example in addition the boundary integral vanishes. After this didactic example let us return to our original discussion. The next step in our derivation is to divide the domain in subdomains the finite elements. Thus we obtain from Eq.(7.2)

To compute an approximate solution of Eq.(7.2) we expand with respect of local approximation functions defined only on the single elements

with the summation over all elements, the label of the basis approximation functions on each element, local coordinates defined locally on each of the elements, see Fig.(7.1), and the expansion coefficients. Thus Eq.(7.5) leads after variation to

Because the expansion coefficients are independent of each other we obtain a linear system of equations to calculate the coefficients

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Thus solving this linear system of equations will approximately solve our original Dirichlet boundary value problem. The derivation above is based on the Ritz-Galerkin principle. In general deriving the correct functional is a non trivial problem. For a system of linear differential equations

a selfadjoint differential operator, given by [4]

a complex function, the functional is

and the corresponding linear system of equations by

with symmetric and banded. After this excursion into the mathematical background of finite elements let us now turn to the quantum applications.

2. 2.1

UNIDIMENSIONAL FINITE ELEMENTS GLOBAL AND LOCAL BASIS

One-dimensional finite elements can be applied either to one degree-offreedom quantum systems or to single components of the multi dimensional Schrödinger equation, e.g, the radial Schrödinger equation. Let us first discuss bound states of a one dimensional quantum system for simplification. Thus the corresponding Schrödinger equation in the coordinate representation is

Due to the simplicity of the kinetic energy operator and the arbitrary complex shape of the potential, finite element techniques are only practicable in the coordinate representation of the quantum system. For simplification we set

Because bound states a left and right border, functions vanishes effectively:

are normalizable we could always find in coordinate space beyond which the wave

By multiplication of the Schrödinger equation with the wave function from the left, we arrive at the equivalent variational equation with

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the functional

Because the wave function vanishes approximately outside the interval we approximate the functional by a finite integration. Integrating by part leads to

Hence we arrive finally at

and with the following preliminary expansion

at

Up to now the basis functions are still arbitrary and not restricted to a finite element approach. In the finite element approximation the space will be divided into small pieces and the wave function will be expanded on each of the elements via interpolation polynomials. On each of this elements we define a local basis, the nodal point coordinates:

with

the length of the n-th finite element,

space of the left-hand border of the element and Fig.(7.1) a simple example is shown.

the global position in the local coordinate. In

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The size of each of the local elements could be optimized individually and could differ between neighboring elements. This freedom allows to refine the finite element grid in those space areas in which the potential shows strong fluctuations and to use a coarser grid in those areas in which the potential behaves rather smoothly. Oversimplification of this loosely spoken statement results in finite elements of constant size for nearly any potential. (That this makes no sense becomes obvious if we compare the wave functions of different systems with each other. Even there asymptotic behavior is quite different.) The task of the finite element is not to optimize the potential shape, but to optimize the computation of the wave functions, which means the eigensolutions of the Schrödinger equation. Thus let us study two examples: the harmonic oscillator wave function and the hydrogen radial wave function. The normalized oscillator eigenfunctions are given as

with the oscillator mass and frequency and the Hermite polynomials to the excited state. Thus the nodes of the eigenfunctions are given by the zeros of the Hermite polynomial. The radial hydrogen eigenfunctions are given by

with Z the charge, the Bohr radius and principal and angular momentum quantum number. Hence for the radial hydrogen eigenfunction the nodes are given by the zeros of the generalized Laguerre polynomials. Hermite and Laguerre polynomials are connected with each other by the square of the function argument, see Chapt. 5.3.4. Already from this simple consideration it would be astonishing if both wave functions are optimized by, up to scaling factors, an identical finite element grid. In Fig.(7.2, top) the nodes of the radial hydrogen eigenfunctions with principal quantum number and are plotted. In contrast to the rather equidistantly distributed nodes of the harmonic oscillator eigenfunctions, Fig.(7.2, bottom), their distribution is roughly quadratically. Thus to optimize the computations the finite element grid will be selected accordingly. Let be the size or length of the finite element and the left border of the element. The finite elements of constant dimension are given by

and the wave functions vanish approximately for As already mentioned above, a quadratic spacing is numerically more favorable for Coulomb

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systems: and hence the size of the elements increases linearly

The wave function on the

finite element is given by

where is the correct value of the wave function at the nodal points and the summation runs over all interpolation polynomials. Hence the values, play the rôle of the expansion coefficients on each of the finite elements and the entire wave function is given by, loosely speaking, putting all pieces together.

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2.2

217

INTERPOLATION POLYNOMIALS

Common interpolation polynomials are the Lagrange, the Hermite and the extended Hermite interpolation polynomials. Unfortunately this interpolation polynomials are often called simply Lagrangian and Hermitian which should not be mixed up with the polynomial functions of the same name. 2.2.1

LAGRANGE INTERPOLATION POLYNOMIALS

Let us first discuss in more detail Lagrange interpolation polynomials, labeled by the index In the following discussion we assume that the whole space of interest is divided in smaller elements carrying a local coordinate system running from [–1, +1]. A simple change of variables will allow us to map this interval onto any other region Derivation from a polynomial expansion. which can be expanded in a Taylor series

Let

be an arbitrary function,

Through any two points there is a unique line, through any three points a unique quadratic and so forth. Thus approximating the function by interpolation polynomials will necessitate polynomials of order if points of the function are given. Approximating function by a polynomial under the following assumption

and

results in

with "1" (in general n) the polynomial degree and the label for the single interpolation polynomials. Thus this interpolation polynomial fulfills

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and the corresponding Lagrange interpolation polynomial and

The general definition of a Lagrange interpolation polynomial of order n is

with equidistant nodes and with

and therefore

Of course Eq.(7.27) could be generalized such that the nodes are non-equidistantly or even randomly distributed in the interval [–1, +1]. For finite element approaches the nodes are usually equidistant in the local coordinates inside the single finite element. An example for the lowest four Lagrange interpolation polynomials is shown in Fig.(7.3). Derivation from a system of linear equations. Instead of the derivation above, Lagrange interpolation polynomials can also be derived from a system of linear equations for the polynomial coefficients, obtained by the polynomials expanded at the nodes This point of view has the advantage that it is straight forward to take in addition derivatives of the original wave function into account. (In the following we will denote all interpolation polynomials by and omit the polynomial order To derive the Lagrange interpolation polynomial, we only assume that our interpolation polynomial fulfills

at the (equidistant) nodes Therefore in local coordinates on a single finite element the wave function is given by

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with correct values at the nodes For any other coordinate values, the wave function will be approximated by the Lagrange interpolation polynomial

Example. A polynomial of order has and hence nodal points in distributed. E.g. for 3 nodes we get hence a system of linear equations

linear independent coefficients which are equidistantly and

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and additional similar equations for and can be condensed to a matrix equation:

This linear system of equations

whose solution will give us the unknown coefficients for the Lagrange interpolation polynomial. The example above can be generalized easily to a polynomial of arbitrary order. Thus we get the following linear equation:

Computing the inverse matrix will lead to the unknown coefficients of the Lagrangian interpolation polynomial. Some results are listed in Tab.(7.1).

2.2.2 HERMITE INTERPOLATION POLYNOMIALS For Lagrange interpolation polynomials the wave function was computed exactly at the nodal points. Nothing could be said about the first derivative of the wave function. For Hermite interpolation polynomials we assume in addition that the first derivative of the wave function will be correctly computed

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at the nodal points. Hence on a finite element we make the following ansatz for the wave function

and for the derivative of the wave function

which leads to

hence nodal points lead to a total of of order

linear equations and thus a polynomial

Example. With we need for two nodes, Hermite interpolation polynomials of order three. Thus for

and we obtain

and similar equations for At the nodal points this will give us the following system of linear equations:

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which could be, in analogy to the derivation of the Lagrangian interpolation polynomials, condensed into a matrix equation

Therefore again the coefficient matrix C will be given by the inverse of the nodal matrix A. Generalization of the equation above is straight forward. Examples of Hermitian interpolation polynomials are shown in Fig.(7.4).

2.2.3

EXTENDED HERMITE INTERPOLATION POLYNOMIALS

The next higher interpolation step are extended Hermite interpolation polynomials in which not only the wave function and its first derivative but in

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addition its second derivative are taken into account. Hence we get

and therefore the polynomials have to fulfill

For nodal points a complete description will be given by coefficients and hence the corresponding polynomial is of the order All further derivations are equivalent to those described in detail for Lagrangian interpolation polynomials and thus left to the reader. An example is shown in Fig.(7.5). Obviously, as defined by Eq.(7.38a), the derivatives of vanishes at all nodal points and the interpolation polynomial respectively at the nodal points.

2.3

EXAMPLE: THE HYDROGEN ATOM

A better understanding how finite elements can be applied to quantum systems is gained by concrete examples. Thus we will study in this subchapter in detail the radial Schrödinger equation of the hydrogen atom. The Schrödinger equation of the hydrogen atom reads

and thus the radial Schrödinger equation in atomic units for vanishing angular momentum

Under the assumption

and partial integration we obtain

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Thus together with Eq.(7.23) this leads to

where in the first line [. . .] is given by Eq.(7.40) and the derivative, e.g. in is with respect to the radial coordinate. In our finite element equation all terms

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should be formulated in the local coordinates. Therefore we have to transform these derivations from the global to the local coordinate:

In Eq.(7.41) we sum over all finite elements. Thus by Eq.(7.41) for each single finite elements the following local matrices are given

By this local approach the computation is significantly simplified, because the structure of the local matrices is independent from the individual finite element and thus the single integrations, e.g.,

has to be done only once for each and hold for any finite element. The contribution from all the elements via local matrices are assembled to construct the global matrices H and S which results finally in a generalized real-symmetric eigenvalue problem

Because S is symmetric and positive definite this can be easily transformed into an ordinary eigenvalue problem. By a Cholesky decomposition we get and thus

Of course we have to bear in mind, that the wave function has to behave smoothly at the element boundary. Which means the local wave function for the element at the right border equals the local wave function at the left border of the neighboring finite element. Thus in the global matrices the local matrices overlap at the border of the elements. An example is presented in Fig.(7.6) for five elements. For Lagrange interpolation polynomials with

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three nodes each block of the Hamiltonian matrix, except the first and the last one, has the following structure

and of course similar the mass or normal matrix S. For Lagrange interpolation polynomials only the first and last matrix element in each matrix block will overlap. Instead of these single block matrix elements for Hermite interpolation polynomials a 2 × 2 sub-block and for extended Hermite interpolation

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polynomials a 3 x 3 sub-block will overlap, because now not only the wave function but in addition the first respectively second derivatives at the borders of the elements have to be equal. 2.3.1 CONVERGENCE As an example we will discuss the convergency of the Coulomb problem under different finite element structures and interpolation polynomials. The dimension of the Hamiltonian matrix and the number of non-vanishing matrix elements does not depend on the physical system under consideration. Because the width of the matrices for Hermite interpolation polynomials are larger the width for Lagrange interpolation polynomials, we will use the number of nonvanishing matrix elements as parameter. Let be the degree-of-freedom of the interpolation polynomials, where this number tells us the number of derivatives of the interpolation polynomial which will be taken into account. Thus

interpolation polynomials. For finite elements with nodes the interpolation polynomial is of the order For elements, the dimension of the banded, symmetric finite element matrices is given by

and the number

of its non-vanishing matrix elements by

In Fig.(7.7) the dependence of the convergence of the energy eigenvalue for the hydrogen eigenstate is presented. Obviously for interpolation polynomials of higher order the energy is much better converged than for low order interpolation polynomials. There is always an interplay between the size of the single finite elements and the order of the interpolation polynomial. Higher interpolation polynomials allow coarser grids. Thus in dependence of the system under consideration one have to find the optimized choice by experience. Higher order polynomials lead to a smaller number of elements, but on the other side the width of matrix bands become larger. The difference between Lagrange and Hermite interpolation polynomials of the same order seems for a fixed number of non-vanishing matrix elements rather small, but nevertheless in all our studies Hermite polynomials are slightly better. Thus by experience, 5th order Hermite interpolation polynomials seem to be best

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for many quantum systems. Increasing the polynomial order beyond 5th order results in some examples in additional wiggles in the wave functions and thus deteriorates the convergence. But the most important point to obtain accurate results is to select a fairly large coordinate space. Outside the finite element area the wave function is equal zero by construction, and thus if this area should be too small the energy will be increased significantly beyond its correct value. One of the advantages of finite element calculations is, that the accuracy of eigenvalues and eigenfunctions are roughly of the same order. Besides the size of the selected space and the choice of the interpolation polynomials the structure of the finite elements decides about the efficiency of the computation. Let us assume that the radial space we have selected is of the order of 100 a.u. and the size of the first element 0.07 a.u. . For finite elements of constant size, this necessitates 1429 elements. For 5th order Hermite interpolation polynomials the dimension of the matrices becomes 5718 and the number of non-vanishing elements For quadratically widened elements only 38 elements are necessary. Thus again for a 5th order Hermite interpolation polynomial, we obtain now finite element matrices of dimension 154 and only non-vanishing matrix elements. This is

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only 2.7% of the value for finite elements of constant size. The only question remains, if we will get convergency. In Fig.(7.8) some results for different finite element structures are shown. Again the hydrogen state was selected and the convergence, measured by the relative deviation of the computational energy E from the exactly known energy as a function of the non-vanishing Hamiltonian matrix elements is presented. The weakest convergence was obtained for finite elements of fixed size, labeled ’const.’. More efficient are finite elements where the interval length increases linearly with the distance from the origin and thus the distance of the border of the finite element from the origin increases quadratically. Even better, but more laborious to obtain the structure, are finite elements which are ’Laguerre-spaced’. These finite elements are selected such, that the size of the n-th finite element in units of the first finite element equals the n-th zero of the Laguerre polynomial in units of the first node. Thus for a total of elements the sizes of the finite elements are given by the zeros of the m-th Laguerre polynomial. In many cases the structure of the physical system is such, that it is not really worth to use adaptive methods. This holds for example

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for the hydrogen atom in external fields, even if the system will run from spherical symmetric to cylindrical symmetric and will become non-integrable and chaotic. Of course the best structure for the selected grid depends on the physical system under consideration. Hydrogen like systems, as presented above, favor quadratical spacing. Adaptive methods become important for multicentre problems.

3.

ADAPTIVE METHODS: SOME REMARKS

The purpose of selfadaptive methods is to seek an optimized grid, such that the error becomes sufficiently small without increasing unnecessarily the finite element space. Global refinements are often inefficient, because the finite element space will be refined even in those areas which do not significantly count to the total error. The strategy of adaptive methods can be summarized as: 1st Evaluate the residual of the current solution on the initial mesh. 2nd Compute the local error, that means the error related to a single finite element. Error computations are based on different norms. For the computation of quantum bound states the so-called energy norm seems most appropriate. 3rd Estimate the global error. 4th Refine the grid due to error analysis. (There are several adaptive finite element solver via Internet charge-free available, e.g., KASKADE [5].) The essential step of grid optimization is the error analysis. For finite elements the total error is based on the restriction of the total space to a subspace. This means, e.g., for bound states the selection of the border at which the wave function is set equal zero. Of course this is part of the formulation of the boundary value problem and hence the mesh construction will have no influence on this error. The second error might occur due to the finite accuracy in computing the integrals related to the local matrices. Again this will not be influenced directly by the grid selection. The third error is related to the mesh size and the order of the interpolation polynomials. There are two possibilities to react on this error: Modifying the grid or changing the order or even type of interpolation polynomials. Selfadaptive methods are based on an analysis of this third error. In the following discussion we will assume that we keep the interpolation polynomial fixed and that we will only modify the mesh, either coarsening or by refining the grid. The error analysis based on the work of Zienkiewicz [4] has become popular for adaptive applications. (For the non-German speaking reader this can also be found in [3], especially Chapt. 14.) Let us start by defining an inner product on finite element space as

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with the domain of our boundary problem, see Eq.(7.l). The defined by

231 is

By an error analysis it could be estimated that the norm of the deviation of the approximated solution from the true solution is proportional to the residual of the computed solution and the size of the finite element to the power of an constant which depends on the smoothness of the exact solution. Thus the only important message at the moment (which seems to be obvious to practitioner) is that the error depends on the selected grid size. In finite element space our quantum system is approximated by the boundary value problem Thus

can be written as

with a lower order operator. For the Laplacian but restricted to Let us define

would be the nabla operator,

Thus in case of the Laplacian would be the gradient of the solution Let us denote the finite element approximation with and let be the normal coordinate of the single finite element. Then the local errors are

(Note, that already and are approximations because is a finite approximation to the infinite coordinate space. If this space will be too small, this approximation will be the main source of the total error which cannot be corrected by the selected mesh.) Of course only the absolute values of the local errors defined above are of interest. To steer the selfadaptive process for quantum systems error estimates employ one of the following two norms1. The energy norm of the error or the flux norm of the error, defined as

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The square of any of these norms is the sum of the corresponding norms over the local finite elements. E.g.

where labels the single finite elements. Thus the remaining problem is that we do not know the exact solution and thus we cannot evaluate the equations above. There are several possibilities to approximate the local errors defined above. Computing the residual of our approximated solution will only tell us how small the total error is, hence can be used to control convergence. One simple method is to refine the mesh globally in the first step, or practically spoken to start with a coarse grid, and to use the new solution as a first order approximation to the correct solution to compute the local error. Another method commonly used is an averaging based on the number and size of elements contributing to a node. For quantum systems we expect and to be globally continuous. Thus a solution should be more accurate for which these values are continuous across element boundaries. Therefore we approximate and by an averaging of and Basis of the adaptively remeshing is then that we want the energy norm of the error to be the same in all the elements and that we want the locally allowed error per element lower a given value. An application of adaptive finite elements based on the software package KASKADE, mentioned above, is presented in [6]. They studied the two-dimensional linear system with triangular finite elements and an adaptive grid generation and obtained results superior to global basis expansions.

4.

B-SPLINES

Splines are quite common in approximating sets of discrete data points by picewise polynomials. For approximations of data sets kubic splines are mainly used. Other applications cover computer-aided design, surface approximations, numerical analysis or partial differential equations. Here we will concentrate on B-splines. For simplicity we will restrict the following discussion onto one-dimensional systems. The first step in using B-splines finite elements is again to confine the quantum system under consideration to a finite space area outside which the wave function effectively vanishes for bound states. This finite area will be divided into single finite elements. In contrast to the interpolation polynomials discussed above, there will be no internal nodes in the single

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finite element. Thus by the endpoints of the finite elements a knot sequence will be defined. The B-splines of order on this knot sequence are defined by the Boor-Cox recursive formula

In particular, is a piecewise polynomial zero outside the interval thus covering finite elements and normalized such that

For a general discussion of splines see, e.g., [7] and for quantum applications, e.g. [8, 9]. To optimize the computation for B-spline finite elements the choice of the grid may be varied in different parts of the space. With respect to grid optimization there is no principle difference between finite elements with interpolation polynomials or finite elements with splines. In contrast to the interpolation polynomials discussed above, B-splines are defined as polynomials piecewise on intervals which are bounded by neighbored grid points. E.g. Lagrange interpolation polynomials of any order will be restricted to one single finite element, but B-splines of oder k will have a non-vanishing contribution for the interval thus covering finite elements. Thus the structure of the global matrices differs significantly

In Eq.(7.56) the show the structure of a global matrix with four finite elements and Lagrangian interpolation polynomials of third order. For third

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order B-spline finite elements2 the global matrices will have additional nonvanishing elements, in Eq.(7.56) marked by Thus comparing the efficiency of B-spline with interpolation polynomials should also take into account the structure of the global matrices and not only the number of nodes, as quite common in literature. B-spline techniques are mainly used for solving the radial Schrödinger equation or a system of coupled uni-dimensional Schrödinger equations. For the radial Schrödinger equation

with ansatz we expand in terms of B-splines. The radial space is divided into finite elements. The endpoints of the finite elements are given by the knot sequence for B-splines of order For the B-splines vanish at their endpoints, except at the first B-spline is equal to one and all others are vanishing and also at the last B-spline has the same behavior. The knot sequence above can in general be distributed arbitrarily in radial space. Hence the expansion of the radial wave function is given by

where the boundary conditions are implemented by restricting the above summation not to include and Due to the boundary condition the wave function has to vanish for and due to the computational assumption for Thus using the Galerkin variational principle the generalized eigenvalue is given by

and

The structure of these symmetric banded matrices is shown in Eq.(7.56). Generalization is possible in two directions: To include resonances and scattering

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states, either the boundary conditions have to be modified or complex coordinate rotations have to be employed. Generalization towards higher dimensions is possible by including tensor B-splines. But the application of B-splines is mainly restricted to uni-dimensional systems.

5.

TWO-DIMENSIONAL FINITE ELEMENTS

For uni-dimensional systems only the size of the finite elements can vary between neighboring elements. In multi-dimensional systems we have in addition the possibility to select different shapes. In two dimensions there are two different shapes common: rectangular and triangular finite elements. The nodes could be located only on the border of the finite element or in addition in the interior. Rectangular finite elements become quadratic in local coordinates. If the two local coordinates of a rectangular finite element are and the interpolation polynomial could be given by a product of two interpolation polynomials or as a function In the first case we could use, e.g., the one-dimensional interpolation polynomials already discussed above. Of course if, e.g., becomes zero the total interpolation polynomial will become zero and if equals one, at each position at which becomes one, the total interpolation polynomial is also one. Thus the total number of nodes is given by the product of the single nodes of and A didactically very well presented application to a two-dimensional quantum system, the hydrogen atom in strong magnetic fields, is [10]. Each polygonal shape can by covered by triangular elements. Even if most potentials in quantum dynamics are not limited by borders we will discuss in detail only triangular finite elements. Using different shaped finite elements is straight forward.

5.1

TRIANGULAR FINITE ELEMENTS

5.1.1 LOCAL COORDINATES In one dimension we have introduced local coordinates such that the size of each finite element becomes one, independent of its true length in space. Thus similar to the uni-dimensional construction we will construct local coordinates for triangular finite elements. Let us start with an arbitrary triangle in global coordinates, see Fig.(7.9), with a point P inside. Connecting this interior point with all edges and labeling these edges counter clockwise, (P1,P2,P3), will separate the triangle in three triangles with area Thus we could define local coordinates by the ratio of the subareas of the triangle with the total area of the triangle F

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Because we started with two dimensions but derived three local coordinates, we need in addition a constraint, which is obviously

These local coordinates have the following properties: and because for the coordinates become in P1 and similar, in and in P3 In Fig.(7.10) the triangular finite element is presented in local coordinates. Of course to obtain the local matrices we have to integrate only over two local coordinates

with {. . .} the integrand and J = 2F the Jacobi determinant and F the total area of the triangle. Again, similar to the uni-dimensional system the contribution from all the elements via local matrices is assembled to construct the

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global matrices, which results finally in a generalized real-symmetric eigenvalue problem. In one dimension the order of the finite element is simple. In two dimensions we have several possibilities to order and hence to label the finite elements. Because common nodes of neighboring elements result in an overlap of the block matrices corresponding to these finite elements, the construction of the finite element net will have an influence on the band width of the matrix. Hence the organization of the finite element grid becomes important. An example is shown in Fig.(7.11). With this construction, e.g., finite element "9" will have common nodes with "3", "10", "11", "18", "17", "16", "15", "14", "7", "8", "1" and "2". Thus it could be even worth to construct different nets in dependence of the selected interpolation polynomials. The finite element grid presented in Fig.(7.11) is efficient for all interpolation polynomials discussed below. 5.1.2 INTERPOLATION POLYNOMIAL In this subsection we will discuss two-dimensional interpolation polynomials useful for quantum computations, All interpolation polynomials are of Lagrangian type. In contrast to engineering problems it turned out, that for most applications, due to the simple shaped quantum potentials, interpolation polynomials with 10 or even 15 nodes are more efficient than using finer grids and linear interpolation polynomials.

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Linear two-dimensional interpolation polynomials. The simplest interpolation polynomials are linear interpolation polynomials. For linear interpolation polynomials the finite element has three nodes located at the corners. An example is shown in Fig.(7.12).

Quadratic two-dimensional interpolation polynomials. For quadratic interpolation polynomials we need six nodes. These nodes are labeled counter clockwise first for the nodes at the three corners and then for the remaining line centered nodes, see Fig.(7.14). An example of quadratic two-dimensional interpolation polynomials is shown in Fig.(7.13). Obviously the interpolation polynomials becomes one at one single node and zero at all other nodes.

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Cubic two-dimensional interpolation polynomials. For third order interpolation polynomials in two dimensions we need finite elements with 10 nodes. 10 nodes cannot be selected uniformly on the border of the triangle, hence one node will be centered inside the triangle. Because interior nodes increase the width of the banded matrices, interior nodes are avoided usually. Again the nodes are labeled counter clockwise, first at the edges of the triangle, then uniformly distributed on the border lines and at last the interior nodes, here of course only one, see Fig.(7.14). An example for cubic interpolation polynomials is shown in Fig.(7.15).

Interpolation polynomials for elements with fifteen nodes. For finite elements with fifteen nodes there are several possibilities to select the interpolation polynomials. In contrast to uni-dimensional finite elements the same number of nodes offers now the possibility to select interpolation polynomials of different order. If all 15 nodes are uniformly distributed on the border of the triangular finite element the interpolation polynomials are of 5th order:

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15 nodes, uniformly distributed on the border will lead to a ‘crowding’ of nodes on the border and to a huge empty area inside the triangular finite element.

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Thus in this case rather uniformly distributed nodes over the whole finite element are more favorable. This can be obtained by fourth order interpolation polynomials. For quartic interpolation polynomials 12 nodes are uniformly distributed on the border and three uniformly in the interior of the triangular finite element. An example for both types of interpolation polynomials with 15 nodes is shown in Fig.(7.16).

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with

5.2

EXAMPLES

The following two examples, the hydrogen atom in strong magnetic fields and the harmonic oscillator, document the principle way how to compute via two dimensional finite elements quantum eigensolutions. The hydrogen atom shows

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in addition the transformation to a different coordinate set as an alternative to varying the size of the finite elements as function of the distance from the origin. 5.2.1 THE HYDROGEN ATOM IN STRONG MAGNETIC FIELDS In [10] eigenenergies of the hydrogen atom in strong magnetic fields computed with rectangular two-dimensional finite elements are discussed. Here we will present a similar type of calculation, but for triangular finite elements. Due to the potential we will use semiparabolic coordinates3 instead of Cartesian or cylindrical coordinates. The Hamilton operator of the hydrogen atom in strong magnetic fields in atomic units reads

with the angular and the spin operator. Due to the cylindrical symmetry will be conserved and thus the angular momentum part will only give rise to a constant energy shift for fixed magnetic quantum number and fixed spin. Due to equs.(2.36) and (2.51) the Schrödinger equation for the hydrogen atom in strong magnetic fields in semiparabolic coordinates becomes

Thus for the kinetic energy we obtain

and for the potential energy

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thus

To compute the local matrices, the equation above has to be evaluated on each of the single elements and the coordinates have to be transformed to local coordinates. Due to the constraint for the local coordinates we arrive finally for the finite element at

and for the local matrix elements

Merging all together we obtain finally the generalized real-symmetric eigenvalue problem (see Eq.(7.44))

whose solution will give us the eigenenergies and corresponding wave functions. Using in each direction 100 elements and 2-dimensional interpolation polynomials of 3rd order results in a 90601 dimensional total matrix, which allows to compute for about 3.5 Tesla the lowest 150 m=0 eigenstates. Of course it is also easily possible to compute eigensolutions for the hydrogen atom in

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stronger magnetic fields. To obtain converged results three parameters have to be checked carefully: the order of the interpolation polynomials, the number of finite elements and the size of the net, which means the border at which the wave function vanishes effectively. 5.2.2 THE TWO-DIMENSIONAL HARMONIC OSCILLATOR The Hamiltonian of the two-dimensional isotropic harmonic oscillator reads for mass

Thus the local matrices become

which are again merged together in the generalized eigenvalue equation

to obtain eigenenergies and eigenfunctions. In Tab.(7.2) some results are presented to document the convergence. By “computation I” (Tab.(7.2) we obtain about 100 eigenstates with an accuracy better 1%. Using a much smaller grid and a much lower number of finite elements (computation II) will result in only 30 computed eigenstates. Using the same size of the net but increasing the number of finite elements, thus refining the grid, will lead to only a few additional eigensolutions and a weakly higher accuracy. Thus computing a higher number of eigensolutions necessitates a larger size of the net. A higher accuracy without increasing the total number of converged eigensolutions can be obtained either by refining the grid or using higher dimensional interpolation polynomials. Two-dimensional finite elements are widely used in quantum computations. One of the early works [11] discusses a comparison between finite element

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methods and spectral methods for bound state problems. A finite element analysis for electron hydrogen scattering can be found in [12]. In many problems the number of degrees-of-freedom is higher two. In those cases a combination of finite element methods with other computational schemes is often more favorable than working with higher dimensional finite elements. [13] used a combination of the closed coupling schemes with finite elements to compute eigenstates for a five degrees-of-freedom system, the helium atom in strong magnetic fields. In the next section we will describe a combination of discrete variable and finite element methods.

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USING DIFFERENT NUMERICAL TECHNIQUES IN COMBINATION

For multi-dimensional systems there is the possibility to combine different numerical techniques to obtain converged solutions. The background is either to obtain an optimized numerical technique or because otherwise storage and/or computational time become too large. The fundamental idea is quite simple. Suppose we have a degrees-of-freedom system. Therefore the wave function will depend on space coordinates. In case of partially separable systems, the wave function will be expanded by a product ansatz

with the separable and the non-separable part. For non-separable systems, respectively the non-separable subsystem, the wave function can be expanded

with the space coordinates and wave functions related to the coordinate splitting. To obtain numerical results different techniques can be applied to each part of the wave function. For systems with three degrees-of-freedom it is quite common to use angular, and radial, coordinates. For systems with cylindrical symmetry the azimuthal angle can be separated off and the magnetic quantum number is conserved. The non-separable part of the Hamiltonian will thus only depend on the angle and the radial coordinate An expansion of the angular part with respect to discrete variables (see Chapt. 5) and the radial part with respect to finite elements [14] turns out to be highly effective and flexible. Because a better understanding in this subject is helped by concrete examples we will discuss as an example alkali metal atoms in strong magnetic fields. The Hamiltonian of alkali metal atoms in strong magnetic fields is given by equation (6.128) with potentials (6.129, 6.130) and the ansatz for the wave function by Eq.(6.133):

Thus the wave function is expanded with respect to the angular and the radial part. For the angular part the Hamiltonian equation and the wave function will be discretized with respect to the nodes of the Legendre polynomial and for cylindrical symmetry symmetrized. The details are described in Chapter (5.4.2.1). This will lead finally to an eigenvalue equation (6.146)

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with a matrix carrying the radial differential part and the centrifugal and angular part. For solving this differential eigenvalue equation we will expand the radial wave function with respect to a finite element ansatz as described in Chapter (6.2.1) and Chapter (6.3). Thus from the equation above we will get finally a generalized eigenvalue equation

with H the global Hamiltonian and S the global mass matrix. The elements of this global matrices are for non-cylindrical systems matrices and for conserved magnetic quantum number matrices due to the angular discretization, with N the number of nodes in the respectively discretization. Because in case of conserved quantum number4 radial wave 5 functions are real the dimension is reduced in addition by a factor of 2. Thus the global matrices have the following structure

with the local matrix element corresponding to the finite element and each of the local matrix elements is again a matrix of dimension respectively N × N. Thus for interpolation polynomials with degrees-offreedom, finite elements each with nodes, and discretization points in the angle, the dimension of these banded matrices is given by

and the number

of its non-vanishing matrix elements by

For the linear algebra routines the important value is the density vanishing matrix elements, which is given by

of non-

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and thus the density of the matrix becomes independent from the degree-offreedom of the interpolation polynomials and from the number of used nodes in the discretization of the angular space via discrete variables. This is one of the reasons why this combination seems so favorable. E.g. for the hydrogen atom in strong magnetic fields of white dwarf stars (100 to 100, 000 Tesla) no convergency could be obtained by using Sturmian functions up to a basis size of 325,000. Such a large basis necessitates supercomputers. For exactly the same problem we could obtain convergent results [14] by a combination of finite elements and discrete variables already on small work stations. Of course this combination is not only restricted to bound states. It can be as well used in combination with complex coordinate rotations to compute resonances or in combination with propagator techniques [15] to compute wave packet propagations. In the complex coordinate method the real configuration space coordinates are transformed by a complex dilatation. The Hamiltonian of the system is thus continued into the complex plane. This has the effect that, according to the boundaries of the representation, complex resonances are uncovered with square-integrable wavefunctions and hence the space boundary conditions remain simple. This square integrability is achieved through an additional exponentially decreasing term

After the coordinates entering the Hamiltonian have been transformed, the Hamiltonian is no longer hermitian and thus can support complex eigenenergies associated with decaying states. The spectrum of a complex-rotated Hamiltonian has the following features [16]: Its bound spectrum remains unchanged, but continuous spectra are rotated about their thresholds into the complex plane by an angle of Resonances are uncovered by the rotated continuum spectra with complex eigenvalues and square-integrable (complex rotated) eigenfunctions. In the Stark effect the whole real energy axis is the continuum spectrum. Therefore no threshold exists for the continuous spectrum to rotate about. The Hamiltonian for the hydrogen atom in parallel electric and magnetic fields, after the above complex transformation has been applied, reads

using atomic units and spherical coordinates. Thus the complex coordinate rotation affects only the radial coordinate but not the angle coordinates and

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hence the discrete variable part remains unchanged. Because the radial coordinate is extended into the complex plane, but, of course, the interpolation polynomials in the finite element ansatz remain real, the expansion coefficients become complex and thus this results in a complex symmetric generalized eigenvalue problem. Fig.(7.17) shows an example for the convergence of a complex eigenvalue as function of the complex angle Convergence to a complex eigenvalue is systematic and follows a pattern with an accumulation point at the correct complex value.

Thus combining different computational strategies opens efficient ways to calculate quantum results. Nevertheless one should be aware of the limits of each method. E.g., for extremely high excited states and magnetic fields of a few Tesla (1 to 10) we got no convergence with discretization techniques but we could compute converged eigenstates (up to about 1200) with a global basis (special Sturmian eigenstates). Thus there is no general computational technique useful for all quantum systems - and even that makes studying the power of computational techniques and developing computational methods so interesting! Developing physical models of real world systems and studying them by numerical methods have to be done with the same care as real experiments.

Notes 1 The third common method is error control with linear functionals.

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2 The definition of the order of B-splines differs from author to author. Thus some authors would define this as a fourth order B-spline. with semi3 The radial coordinate is given by parabolic coordinates. Thus we get with linearly spaced finite elements in semiparabolic coordinates quadratically spacing in 4 The quantum number is conserved for systems with cylindrical symmetry. 5 In this example the only imaginary part in the Hamiltonian comes from the linear Zeeman term. Of course for other quantum systems this might not be true and thus the radial wave function might remain complex.

References [1] Nordholm, S., and Bacskay, G. (1976). “Generalized finite element method applied to the calculation of continuum states,” Chem. Phys. Lett. 42, 259 – 263 [2] Searles, D. J., and von Nagy-Felsobuki, E. I. (1988). “Numerical experiments in quantum Physics: Finite-element method,” Am. J. Phys. 56, 444 – 448 [3] Akin, J. E. (1998). Finite Elements for Analaysis and Design, Academic Press, London [4] Zienkiewicz, O. C. (1984) Methode der Finiten Elemente, Carl Hanser Verlag, München [5] Beck, R., Erdmann, B., and Roitzsch, R. (1995). KASKADE 3.0 An objectoriented adaptive finite element code Technical Report TR 95-4, KonradZuse-Zentrum, Berlin [6] Ackermann, J., and Roitzsch, R. (1993). “A two-dimensional multilevel adaptive finite element method for the time-independent Schrödinger equation,” Chem. Phys. Lett. 214, 109 – 117 [7] deBoor, C. (1978) A practical Guide to Splines Springer-Verlag, Heidelberg [8] Sapirstein, J. and Johnson, W. R. (1996). “The use of basic splines in theoretical atomic physics,” J. Phys. B 29, 5213 – 5225 [9] Johnson, W. R., Blundell, S. A., and Sapirstein, J. (1988). “Finite basis sets for the Dirac equation constructed from B splines,” Phys. Rev. A 37, 307 – 315

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[10] Ram-Mohan, L. R., Saigal, S., Dossa, D., and Shertzer, J. (1990). “The finite-element method for energy eigenvalues of quantum mechanical systems,” Comp. Phys. Jan/Feb, 50 – 59 [11] Duff, M., Rabitz, H., Askar, A., Cakmak, A., and Ablowitz, M. (1980). “A comparison between finite element methods and spectral methods as applied to bound state problems,” J. Chem. Phys. 72, 1543 – 1559 [12] Shertzer, J., and Botero, J. (1994). “Finite-element analysis of electronhydrogen scattring,” Phys. Rev. A 49, 3673 – 3679 [13] Braun, M., Schweizer, W., and Herold, H. (1993). “Finite-element calculations for S-states of helium,” Phys. Rev. A 48, 1916 – 1920; Braun, M., Elster, H., and Schweizer, W. (1998). “Hyperspherical close coupling calculations for helium in a strong magnetic field,” ibid. 57, 3739 – 3744 [14] Schweizer, W. and Faßbinder, P., (1997). “Discrete variable method for nonintegrable quantum systems”, Comp. in Phys. 11, 641–646 [15] Faßbinder, P., Schweizer, W. and Uzer, T (1997). “Numerical simulation of electronic wavepacket evolution,” Phys. Rev. A 56, 3626 – 3633 [16] Ho, Y. K. (1983). “The method of complex coordinate rotation and its applications to atomic collision processes,” Phys. Rep. 99, 1 – 68

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Chapter 8 SOFTWARE SOURCES

The first step in computational physics is to derive a mathematical model of the real world system we want to describe. Of course this system will be limited by the numerical methods known and by the computer facility available. Many tasks in computing are based on standard library routines and well known methods. Thus it is a question of scientific economy to use already available codes respectively to modify those codes for our own needs. The following list is of course an incomplete list of both, free available and commercial software, but might be nevertheless useful.

General sources for information GAMS. is a data bank in which the subroutines of the most important mathematical libraries are documented. Gams is an abbreviation for “Guide to available Mathematical Software” and can be found under http://gams.nist.gov NETLIB. is a source for mathematical software, papers, list of addresses and other mathematically related information under http://www.netlib.org Physics Database and Libraries. Links and a list of data of special interest for physicists can be found under http://www.tu.utwent.nl/links/physdata.html Numerical and Computer Data Base. Links to and a list of numerical and computer related resources can be found under http://www.met.utah.edu/ksassen/homepages/zwang/computers.html 255

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ELIB. is a numerically related data bank from the Konrad-Zuse Centre (Berlin) http://elib.zib.de MATH-NET. is an information service for mathematicians http://www.math-net.de Linux. a list of scientific applications under Linux could be found here http://sal.kachinatech.com.sall.shtml and in addition Linux software for scientists and other useful information under http://www-ocean.tamu.edu/min/wesites/linuxlist.html GOOGLE. general non-specialized information http://www.google.com

Commercial Software MATLAB. is a numerical software system for solving scientific and engineering problems. It provides advanced numerical codes, including sparse matrix computations based on Krylov-techniques as well as advanced graphical methods. Matlab is the basic package which allows to solve problems based on algebraic, optimization, ordinary differential equations and partial differential equations in one space and time variable, and allows to integrate FORTRAN, C and C++ code. The basic functionality can be extended by additional toolboxes. Some are of interest for physicists, e.g.: “the optimization toolbox”, “the statistical toolbox”, “the partial differential equation toolbox”, “the spline toolbox” and “simulink” for the simulation of dynamical systems, and “the symbolic math toolbox” and “the extended symbolic math toolbox” which allow symbolic computations based on a MAPLE kernel. In addition there are toolboxes for developing stand-alone applications: “the matlab compiler”, “the matlab C/C++ math-library” and “the matlab C/C++ graphics library”. http://www.mathworks.com FEMLAB. is a numerical software system based on Matlab for solving partial differential equations via finite elements. http://www.femlab.com MAPLE. is a symbolic programming system with symbolic, numerical and graphical possibilities. http://www.maplesoft.com

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MATHEMATICA. like MAPLE Mathematica is a symbolic programming system with symbolic, numerical and graphical possibilities. There are several additional libraries available. http://www.mathematica.com Axiom.

http://www.nag.com

Derive. is one of the smallest computer algebra systems and can already be installed on small computers. http://www.derive.com Macsyma.

http://www.macsyma.com

MATHCAD. offers symbolic and numerical computations. C/C++ code could be integrated and graphical functions are available. http://www.mathsoft.com MUPAD. http://www.uni-paderborn.de/MuPAD NAG. in contrast to the software listed above, NAG does not offer its own computational language. NAG is an extensive collection of library subroutines in FORTRAN 77/90 and C. Between Matlab and NAG there is a connection via the “NAG foundation toolbox”. NAG offers more than 1000 routines covering a wide area of numerical mathematics and statistics and is one of the “standard libraries” for physical applications, http://www.nag.com IMSL. consists of more than 500 routines written in C or FORTRAN and covers numerical and statistical methods. http://www.vni.com and http://www.visual-numerics.de Numerical Recipes. The numerical recipes (Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes Cambridge University Press Cambridge) are a well-known book for different numerical methods. Source codes are available on CD in C, Fortran 77 and Fortran 90. http://www.nr.com

Public Domain Software BLAS. is an abbreviation for “Basic Linear Algebra Subroutines” and offers linear algebra basic routines. Blas was the first implemented in 1979. In the meantime 3 levels are available. Blas-level-1 are elementary vector operations

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like scalar products. This are vector-vector operations, respectively operations. For vector computers these routines are not sufficient. Therefore Blas-2-codes were developed. These codes support matrix-vector computations, respectively Because level-2 routines are insufficient for workstations based on risc processors Blas-3 routines were developed. (Lapack is based on Blas-level-3 routines.) http://www.netlib.org/blas EISPACK. program codes for eigenvalue problems http://www.netlib.org/eispack http://www.scd.ucar.edu/softlib/eispack.html Many routines published in the eispack library are based on Algol routines discussed in Wilkinson, J. H. (1965). The Algebraic Eigenvalue Problem Oxford University Press, New York. FFTPACK. programm codes for fast Fourier transformation http://www.netlib.org/fftpack FISHPACK. program codes for solving the Laplace-, Poisson- and Helmholtzequation in two dimensions. http://www.netlib.org/fishpack HOMEPACK.

program codes for solving non-linear systems of equations

ITPACK. software with iterative solvers for large linear systems of equations with sparse matrices. http://www.netlib.org/itpack LAPACK. stands for linear algebra package and consists of FORTRAN 77 routines for linear problems. Lapack covers all standard linear problems and offers additional routines for banded and sparse matrices. Lapack is the successor of linpack and eispack and most routines are more robust than the former ones. In addition there is a parallel version of Lapack available and a C/C++ and Java version. http://www.netlib.org/lapack LINPACK. codes for solving linear systems of equations. The successor of linpack is lapack. http://www.netlib.org/linpack MINPACK. codes for non-linear systems of equations and optimization routines. http://www.netlib.org/minpack

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ODEPACK. software package for solving initial value problems for ordinary differential equations. http://www.netlib.org/odepack ODRPACK. FORTRAN subroutines for solving orthogonal approximations and least square fits. http://www.netlib.org/odrpack QUADPACK. routines for the computation of integrals and integral transformations. http://www.netlib.org/quadpack OCTAVE. numerical computations, but restricted graphical possibilities. http://www/che.wisc.edu/octace/ CERN. Information of special interest for physicists and a numerical library is available from CERN. CERN offers also a “Physical Analysis Workstation” under the abbreviation “PAW” software, paw http://www.info.cern.ch/asd ARPACK. A library for lanczos and arnoldi routines can be found under http://www.caam.rice.edu/software/arpack PORT. this mathematic library covers many numerical applications from the bell laboratories and is free availbale for non-commerical applications for scientists. http://www.bell-labs.org/liblist.html

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Acknowledgments

Many people have been involved, directly and indirectly, throughout the progress of this book, and their contributions were invaluable to me. Earning my money in industry and teaching at the university Tübingen only unsalaried, the book was written during my “leisure time”. Therefore I have discovered as many authors before - that the acknowledgment to the author’s family, often expressed in an apologetic tone, cannot be disqualified as a cliché. My biggest debt is to my wife Ursula for her patience and perpetual support during the long, and sometimes difficult time writing this book. My thanks to the Ph.D. students I supervised: Peter Faßbinder, Rosario González-Férez, Wilfried Jans, Matthias Klews, Roland Niemeier, Michael Schaich, Ingo Seipp, Matthias Stehle, Christoph Stelzer. Each one has taught me some aspects of life in general and contributed in their own way to my intuition and to the topics discussed in the book. Special thanks are due to Matthias Klews and Rosario González-Férez for a critical reading and their helpful comments and criticisms. Of course, it is customary to thank someone for doing an excellent job of typing. I only wish I could do so. It was also a pleasure for me to teach and to learn from many students which took my courses or made their one-year diploma thesis under my supervision. I hope it is clear that a book does not grow in a vacuum. I have taken help from any book or publication that I myself have found worthwhile; those especially useful in writing a chapter have been referenced at the end of that chapter. Thus I apologize for not citing all contributions to the body which are worth reading. It is a pleasure to thank the many colleagues and friends from whom I learnt so much during the past two decades. Special thanks to my supervisor Peter Kramer who teached me the beauty of symmetry structures in physics, Hanns Ruder whose enthusiasm makes him a perfect instructor, and to Turgay Uzer, Günter Wunner, Moritz Braun, Harald Friedrich, Stefan Jordan, Pat O’Mahony, Vladimir Melezhik, Jesús Sánchez-Dehesa, Peter Schmelcher, and 261

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Ken Taylor for the constant encouragement and valuable discussions, and to Susanne Friedrich and Roland Östreicher. Without their help I would not have made the unique experience of observing white dwarf stars with the 3.5-mtelescope at Calar Alto (Spain), and I would not have had the possibility to compare my own computations based on discrete variable techniques and finite elements with spectroscopic data from my own observations. I well-come comments and suggestions from readers. Throughout the book I chose to use the first person plural. Thus who is this anonymous ghostwriter - who are we ? We are the author and the reader together. It is my hope that the reader will follow me when we consider a numerical technique, a derivation, or when we show that something is true. In this way we make contributions towards a better understanding of computational physics.

Index

1/N-shift expansion, 70 3j-symbol, 11, 14 Absorption, 91 Adam-Bashforth algorithm 4th order, 139 Adam-Moulton algorithm 4th order, 139 Adams-Bashforth algorithm, 139 Adams-Moulton algorithm, 139 Aitken’s process, 59 Alkali atoms, 187–188 external fields, 187 Alkali-like ions, 188 Angular coordinates, 162 Angular momentum, 9 coupling, 10 SO(4, 2, R), 79 Anharmonic oscillator, 182 1st order perturbation, 183 DVR, 183 parity, 184 Antisymmetrizer, 106 Associated Legendre differential equation, 170 Associated Legendre function, 10, 161, 168 Avoided crossing, 37, 62 Azimuthal symmetry, 161 Bessel function, 176 ordinary, 176 Rodrigues formula, 177 spherical, 176 Blas, 258 Bohr N., 2 Bohr-Sommerfeld quantization, 2 Boor-Cox recursive formula, 233 Born-Oppenheimer approximation, 46, 111 Boundary condition Dirichlet, 210 generalized von Neumann, 210 Boundary value problem, 210 Bound states, 6

Bra, 8 B-splines, 232 Campbell-Baker-Hausdorff theorem, 18 Cayley approximation, 17 Cayley method, 141 Chaos, 35 Chebyshev polynomial, 182 Christoffel-Darbaux relation, 158 Christoffel-Darboux relation, 167 Clebsch-Gordan coefficient, 10 Commutator, 4 Complex coordinate method, 6, 250 Complex coordinate rotation, 7, 164, 250 Confluent hypergeometric equation, 181 Confluent hypergeometric function, 181 Constant of motion, 35 Contact potential, 2 Continuous states, 6 Coordinates Cartesian, 39 cone, 45 cylindrical, 42 ellipsoidal, 44 elliptic, 46 elliptic cylindrical, 44 hyperspherical, 48 Jacobi, 48 oblate spheroidal, 44 parabolic, 42 paraboloidal, 44 polar, 48 prolate spheroidal, 44 semiparabolic, 45 semiparabolic cylindrical, 44 spherical, 40 Coulomb integral, 115 Coulomb systems, 187 Crank-Nicholson approximation, 17 De Broglie L.-V., 1

263

264

NUMERICAL QUANTUM DYNAMICS

De Broglie relation, 2 Defolded spectrum, 38 Degenerate perturbation theory, 60 Density functional theory, 95, 106 Density matrix, 21 Density operator, 20–21 Dipole strength, 193 Direct product basis, 160 Dirichlet boundary condition, 210 Discrete variable representation, 156, 159 fixed-node boundary condition, 196, 198 hermite polynomial, 183 Legendre polynomial, 189 cylindrical symmetry, 191 periodic, 196, 199 spectroscopic quantities, 193 time-dependent systems, 195 wave packet propagation, 195 DVR, 159 Dynamical group, 78 Dynamical Lie algebra, 78 Dyson series, 16 Ehrenfest’s theorem, 20 Ehrenfest’s Theorem, 20 Einstein A., 1 Eispack, 258 Energy-time uncertainty relation, 89 Euler angles, 11 Euler method explicit, 134 implicit, 134 Euler methods, 134 Euler theorem, 109 Exchange integral, 115 Expectation value, 4 FBR, 159 Fermion problem, 125 Fermi’s Golden Rule, 90 Feynman path integral, 126 Finite basis representation, 156, 159 Finite differences bound states, 146 bound states Pöschl-Teller potential, 147 Hamiltonian action, 140 propagation, 140 potential wall, 146 Finite element matrix dimension, 227 non-vanishing elements, 227 Finite elements, 209, 211 adaptive methods, 230 error, 230 rectangular, 235 selfadaptive methods, 230 splines, 232 triangular, 235

local coordinates, 235 two-dimensional, 235 Fixed-node method, 125 Fractional charge, 43 Free electron magnetic field, 203 Galerkin principle, 212 Gauss-Chebyshev quadrature 1st kind, 199 2nd kind, 198 Gaussian orthogonal ensemble, 68 Gauss quadrature, 167 Gauss-Symplectic-Ensemble, 68 Gauss theorem, 181 Gauss-Unitary-Ensemble, 68 Gear algorithm, 139 Gegenbauer polynomials, 177 generating function, 177 Rodrigues formula, 178 Generalized Laguerre polynomial, 171 Generating function, 167 Guiding function, 124 Hahn polynomials, 180 Hamiltonian, 6 Harmonic oscillator, 182 Hartree equation, 105 Hartree-Fock equation, 106 Hartree-Fock method, 95 Hartree-Fock method, 104 Heisenberg uncertainty, 89 Heisenberg uncertainty relation, 4 Hellman-Feynman theorem, 109 Hermite interpolation polynomial, 220 extended, 222 Hermite mesh, 200 Hermite polynomial, 174 finite elements, 215 generating function, 174 Jacobi matrix, 175 parity, 175 Rodrigues formula, 174 Homogeneous function, 108 Husimi function, 23 Hydrogen atom electric field, 43 external fields, 62 extremely strong magnetic field, 203 finite elements two-dimensional, 244 strong external fields, 187 Hydrogen eigenstates boson representation, 79 Hydrogen molecule-ion, 46 Hylleraas ansatz helium, 99 Hypergeometric functions, 180 Hypergeometric series, 180

Index Hypervirial theorem, 111 Imsl, 257 Inner product, 8 Integrability, 35 Interpolation polynomial degree of freedom, 227 two-dimensional cubic, 239 linear, 237 quadratic, 238 quartic, 241 quintic, 240 Jacobi matrix, 167 Jacobi polynomial, 14, 181 generating function, 182 Kepler problem, 49 Ket, 8 Kicked systems, 141 Kohn-Sham equations, 107 Krylov space, 103 Kustaanheimo-Stiefel coordinates, 51–52 Kustaanheimo-Stiefel transformation, 51–52 Lagrange interpolation polynomial, 217 linear equations, 218 n-th order, 218 Laguerre mesh, 200–201 modified, 202 Laguerre polynomial, 171, 175 finite elements, 215 generalized, 171 generating function, 172 Jacobi matrix, 172 Rodrigues formula, 171 Lanczos method, 101 algorithm, 102 spectrum transformation, 101 Lanczos software, 259 Landau function, 203 Lapack, 258 Laplace-Beltrami operator, 39 cylindrical coordinates, 42 elliptic coordinates, 47 parabolic coordinates, 42 polar coordinates, 51 semiparabolic coordinate, 45 spherical coordinates, 40 LDA, 107 Leap-frog algorithm, 135 Legendre differential equation, 170 Legendre function associated, 168 parity, 171 Legendre polynomial, 168, 187 generating function, 171 Jacobi matrix, 170 Rodrigues formula, 168

Legendre polynomials, 161 Level repulsion, 37 Lie algebra dynamical, 78 spectral generating, 78 Liouville equation, 22 Local basis, 213 local matrices, 225 Local density approximation, 107 Maple, 256 Mathematica, 257 Matlab, 256 Mesh calculation, 200 Mesh Laguerre, 200 Metropolis algorithm, 121 Model potentials, 188 Monte Carlo method, 118 important sampling, 121, 124 quantum, 118 diffusion, 123 guiding function, 124 path integral, 128 variational, 122 Moyal brackets, 24 Müller method, 179 Murphy equation, 170 Nag, 257 Nearest neighbor distribution, 37, 65 Neumann series, 16 Newton’s method, 179 Noether’s theorem, 36 Non-integrable systems, 35 N-particle systems, 48 Numerov integration, 164 Numerov method, 150–151 eigenvalue problem, 151 Ordinary Bessel function, 176 Orthogonal polynomials, 165 three-term recurrence relation, 166 zeros, 166 Orthonormal polynomials, 165 Oscillator anharmonic, 182 harmonic, 182 Oscillator strength, 193 Partial differential equation elliptic, 3 hyperbolic, 3 parabolic, 3 Particle-in-box eigenfunction, 198 Perturbation theory, 55 degenerate, 60 Rayleigh-Schrödinger, 56 two-fold degenerate, 61 Phase space, 35 Photoelectric effect, 2

265

266

NUMERICAL QUANTUM DYNAMICS

Photoelectron, 2 Photons, 2 Planck constant, 2 Planck-Einstein relations, 2 Planck M., 1 Pochhammer symbol, 180 Poisson bracket, 35 Poisson distribution, 38, 65, 120 Polar coordinate, 51 Polarization, 194 194 194 194 Pollaczek polynomials, 180 Pöschl-Teller potential, 147 Predictor-corrector method, 139 Probability conservation, 5 Probability current density, 5 Probability density, 4 Probability distribution, 120 QR-decomposition, 103, 178 Quadratic Stark effect, 69 Quantum force, 124 Quantum integrability, 36 Quantum Monte Carlo, 95, 118 Quantum number good, 36 Quantum potential, 25 Quasibound state, 6 Random number, 119 quasi, 120 Rayleigh-Ritz variational principle, 96 Rayleigh-Schrödinger perturbation theory, 56 Regge symbol, 11 Regge symbols, 194 Released-node method, 125 Resonance, 6 Ritz-Galerkin principle, 212 Ritz variational principle, 210 Rodrigues formula, 167 Romberg integration, 119 Runge-Kutta algorithm, 135 Runge-Kutta-Fehlberg algorithm, 137 Runge-Lenz operator, 79 Scalar product, 8 Schrödinger equation, 2 Schrödinger equation radial, 41 Schrödinger equation time-dependent, 3 Schrödinger equation time-independent, 5 Schwinger’s action principle, 126 Secant method, 179 Self-consistent field, 105 Self-consistent potential, 105 Separability

Cartesian coordinates, 39 cylindrical coordinates, 42 elliptic coordinates, 47 parabolic coordinates, 42 semiparabolic coordinates, 45 spherical coordinates, 40 Separation of variables, 35 Simulink, 256 Slater determinant, 106 Softare netlib, 255 Software arpack, 259 axiom, 257 bell-labs, 259 blas, 258 cern, 259 derive, 257 differential equations, 259 eigenvalue problems, 258 eispack, 258 elib, 256 femlab, 256 FFT, 258 fftpack, 258 fishpack, 258 gams, 255 google, 256 Helmholtz equation, 258 homepack, 258 imsl, 257 itpack, 258 lanczos, 259 lapack, 258 Laplace equation, 258 linpack, 258 macsyma, 257 maple, 256 mathcad, 257 mathematica, 257 math-net, 256 matlab, 256 minpack, 258 mupad, 257 nag, 257 numerical recipes, 257 octave, 259 odepack, 259 odrpack, 259 paw, 259 Poisson equation, 258 port, 259 quadpack, 259 simulink, 256 Solver explicit, 133 implicit, 133

Index multi-step, 133 single-step, 133 Spectral generating group, 78 Spectral representation, 156–157, 159 Spherical Bessel function, 176 Spherical coordinates, 161 Spherical harmonics, 10, 161 Spin, 10 Splines, 232 Stark effect, 43, 68 quadratic, 69 Stimulated emission, 91 Sum rule, 69 Symmetry group, 78 Time discretization, 141 pulse, 141 Time evolution operator, 15 Time propagator, 15 Transformation active, 11 passive, 11 Transition group, 78 Transition probability, 87–88, 193 Trapezoidal rule, 118 Trotter number, 129 Ultraspherical polynomials, 177

Uncertainty relation, 4 energy time, 89 Variance, 4 Variational method diagonalization, 100 harmonic oscillator, 97 helium ground state, 98 Variational principle, 95 Rayleigh-Ritz, 96 Variation basis representation, 156–157 Verlet algorithm, 135 Virial theorem, 108, 111, 117 molecules, 113 variational ansatz, 117 Von Neumann boundary condition, 210 generalized, 210 Von Neumann equation, 21–22 Walker, 121 Wave packet propagation, 195 Weight function, 120, 165 moment, 165 Weyl transformation, 23 Wigner distribution, 38 Wigner function, 20, 22, 68 Wigner rotation function, 13 Wigner-Seitz radius, 108 WKB approximation, 24 momentum space, 26

267