QUANTUM MECHANICS USING COMPUTER ALGEBRA

Transcription

1 QUANTUM MECHANICS USING COMPUTER ALGEBRA Includes Sample Programs in C++, SymbolicC++, Maxima, Maple, and Mathematica 2nd Edition

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3 QUANTUM MECHANICS USING COMPUTER ALGEBRA Includes Sample Programs in C++, SymbolicC++, Maxima, Maple, and Mathematica 2nd Edition Willi-Hans Steeb University of Johannesburg, South Africa Yorick Hardy University of Johannesburg, South Africa World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. QUANTUM MECHANICS USING COMPUTER ALGEBRA Includes Sample Programs in C++, SymbolicC++, Maxima, Maple, and Mathematica (2nd Edition) Copyright 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN ISBN Printed in Singapore.

5 Preface Solutions to problems in quantum mechanics are important for scientists, engineers and many others. This book gives a collection of most standard methods in quantum mechanics together with their programs in SymbolicC++, Maxima, Mathematica and Maple. Advanced topics in quantum mechanics are also included. In most cases the output of the programs is also displayed. Most of the problems are implement in SymbolicC++ and Maxima. For a number of selected problems the program are implemented in Mathematica, Maple and C++. In the first edition the programs had been implemented in Reduce. SymbolicC++, Maxima, Reduce, Mathematica and Maple are the most widely available and simple to use computer algebra systems. They enable users to manipulate algebraic expressions and equations symbolically. For example, we can differentiate and integrate symbolically. Number crunching can also be done. Moreover, symbolic manipulation and number crunching can be combined in one program. Beside the standard methods, modern developments in quantum mechanics are also included. These include Bose operators, Fermi operators, coherent states, squeezed state, gauge theory, quantum groups and super Lie algebras. All the special functions (such as Hermite, Chebyshev, Legendre) important in quantum theory and Hilbert space theory are also implemented. The level of presentation is such that one can study the subject early on in ones education in science. There is a balance between practical computation and the underlying mathematical theory. The book is ideally suited for use in a quantum mechanics lecture. The web sites for the different packages are: v

6 vi Preface Maxima: SymbolicC++: Reduce: Maple: Mathematica: Without doubt, this book can be extended. If you have comments or suggestions, we would be pleased to have them. The addresses of the authors are:

7 Contents Preface v 1 Introduction 1 2 Conservation Law and Schrödinger Equation 4 3 Wave Packet and Free Schrödinger Equation 6 4 Separation Ansatz and Schrödinger Equation 8 5 Matrix Representation in the Hilbert Space L 2 [ π, π] 10 6 One-Dimensional Potential and Trial Function 12 7 Heisenberg Equation of Motion 14 8 Variance 16 9 Unitary Operators Unitary and Hermitian Operators Magnus Expansion Quantum Harmonic Oscillator Harmonic Oscillator and Recursion Relation Commutation Relations of ˆp andˆq Wigner Characteristic Functions Anharmonic Oscillator 34 vii

8 viii Quantum Mechanics using Computer Algebra 17 Morse Potential and Lie Algebra so(2, 1) One-Dimensional WKB-Solutions Angular Momentum Operators I Angular Momentum Operators II Angular Momentum Operators III Lie Algebra su(3) and Commutation Relations Spin-1 Lie Algebra and Commutation Relations Radial Symmetric Potential and Bound States Wave Function of Hydrogen Atom I Wave Function of Hydrogen Atom II Two-Body Problem Helium Atom and Trial Function Stark Effect Scattering in One-Dimension Gauge Theory Driven Two Level System Berry Phase Free Electron Spin Resonance Two-Point Ising-Model with External Field Two-Point Heisenberg Model Spectra of Small Spin Clusters Fermi Operators Fermi Operators with Spin and the Hubbard Model Bose Operators 96

9 Contents ix 41 Bose Operators and Number States Matrix Representation of Bose Operators Quartic Hamilton Operator and Bose Operators Coherent States Squeezed States Bose-Fermi Systems Dirac Equation and Dispersion Law Perturbation Theory Elastic Scattering Entanglement I Entanglement II Teleportation Exceptional Points Expansion of exp(l)a exp( L) Expansion of (A ɛb) Heavyside Function and Delta Function Legendre Polynomials Associated Legendre Polynomials Laguerre Polynomials Hermite Polynomials Chebyshev Polynomials Airy Functions Spherical Harmonics Clebsch-Gordan Series 162

10 x Quantum Mechanics using Computer Algebra 65 Hypergeometric Functions Eigenvalues and Hypergeometric Differential Equation Gamma Matrices and Spin Matrices Hilbert Space and Fourier Expansion Continuous Fourier Transform Plancherel Theorem Wavelets and Hilbert Space Group Theory Permutation Groups and Permutation Matrices Reducible and Irreducible Representations Pauli Group and Clifford Group Lie Groups Quantum Groups Lie Algebras Super-Lie Algebra Casimir Operator and Lie Algebras Gram-Schmidt Orthogonalisation Process Soliton Theory and Quantum Mechanics Padé Approximation Cumulant Expansion Kronecker and Tensor Product 225 Bibliography 229 Index 233