Quantum Mechanics Using Computer Algebra

Includes Sample Programs in C++, SymbolicC++, Maxima, Maple, and Mathematica

Willi-Hans Steeb, Yorick Hardy;;;

• Publisher: WSPC
• Year: 2010
• eBook ISBN: 9789813107892
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Programming

Solving problems in quantum mechanics is an essential skill and research activity for physicists, mathematicians, engineers and others. Nowadays, the labor of scientific computation has been greatly eased by the advent of computer algebra packages, which do not merely perform number crunching, but also enable users to manipulate algebraic expressions and equations symbolically. For example, the manipulations of noncommutative operators, differentiation and integration can now be carried out algebraically by the computer algebra package.

This book collects standard and advanced methods in quantum mechanics and implements them using SymbolicC++ and Maxima, two popular computer algebra packages. Throughout, the sample programs and their outputs are accompanied with explanatory text of the underlying mathematics and physics explained in detail. Selected problems have also been implemented using two other popular packages — Mathematica and Maple — while some problems are implemented in C++.

Modern developments in quantum theory are covered extensively, beyond the standard quantum mechanical techniques. The new research topics added to this second edition are: entanglement, teleportation, Berry phase, Morse oscillator, Magnus expansion, wavelets, Pauli and Clifford groups, coupled Bose–Fermi systems, super-Lie algebras, etc.

Table of contents

• Contents
• Preface
• 1 Introduction
• 2 Conservation Law and Schr¨odinger Equation
• 3 Wave Packet and Free Schr¨odinger Equation
• 4 Separation Ansatz and Schr¨odinger Equation
• 5 Matrix Representation in the Hilbert Space L2[.π, π]
• 6 One-Dimensional Potential and Trial Function
• 7 Heisenberg Equation of Motion
• 8 Variance
• 9 UnitaryOperators
• 10 Unitary and Hermitian Operators
• 11 Magnus Expansion
• 12 Quantum Harmonic Oscillator
• 13 Harmonic Oscillator and Recursion Relation
• 14 Commutation Relations of p and q
• 15 Wigner Characteristic Functions
• 16 Anharmonic Oscillator
• 17 Morse Potential and Lie Algebra so(2, 1)
• 18 One-Dimensional WKB-Solutions
• 19 Angular Momentum Operators I
• 20 Angular Momentum Operators II
• 21 Angular Momentum Operators III
• 22 Lie Algebra su(3) and Commutation Relations
• 23 Spin-1 Lie Algebra and Commutation Relations
• 24 Radial Symmetric Potential and Bound States
• 25 Wave Function of Hydrogen Atom I
• 26 Wave Function of Hydrogen Atom II
• 27 Two-Body Problem
• 28 Helium Atom and Trial Function
• 29 Stark Effect
• 30 Scattering in One-Dimension
• 31 Gauge Theory
• 32 Driven Two Level System
• 33 Berry Phase
• 34 Free Electron Spin Resonance
• 35 Two-Point Ising-Model with External Field
• 36 Two-Point Heisenberg Model
• 37 Spectra of Small Spin Clusters
• 38 Fermi Operators
• 39 Fermi Operators with Spin and the Hubbard Model
• 40 Bose Operators
• 41 Bose Operators and Number States
• 42 Matrix Representation of Bose Operators
• 43 Quartic Hamilton Operator and Bose Operators
• 44 Coherent States
• 45 Squeezed States
• 46 Bose-Fermi Systems
• 47 Dirac Equation and Dispersion Law
• 48 Perturbation Theory
• 49 Elastic Scattering
• 50 Entanglement I
• 51 Entanglement II
• 52 Teleportation
• 53 Exceptional Points
• 54 Expansion of exp(L)Aexp(-L)
• 55 Expansion of (A - B)-1
• 56 Heavyside Function and Delta Function
• 57 Legendre Polynomials
• 58 Associated Legendre Polynomials
• 59 Laguerre Polynomials
• 60 Hermite Polynomials
• 61 Chebyshev Polynomials
• 62 Airy Functions
• 63 Spherical Harmonics
• 64 Clebsch-Gordan Series
• 65 Hypergeometric Functions
• 66 Eigenvalues and Hypergeometric Differential Equation
• 67 Gamma Matrices and Spin Matrices
• 68 Hilbert Space and Fourier Expansion
• 69 Continuous Fourier Transform
• 70 Plancherel Theorem
• 71 Wavelets and Hilbert Space
• 72 Group Theory
• 73 Permutation Groups and Permutation Matrices
• 74 Reducible and Irreducible Representations
• 75 Pauli Group and Clifford Group
• 76 Lie Groups
• 77 Quantum Groups
• 78 Lie Algebras
• 79 Super-Lie Algebra
• 80 Casimir Operator and Lie Algebras
• 81 Gram-Schmidt Orthogonalisation Process
• 82 Soliton Theory and Quantum Mechanics
• 83 Pad´e Approximation
• 84 Cumulant Expansion
• 85 Kronecker and Tensor Product
• Bibliography
• Index