Quantum Methods With Mathematica

The first quantum mechanics text published that ties directly into a computer algebra system, this book exploits Mathematica(r) throughout for symbolic, numeric, and graphical computing. It is a work designed for computer interaction in an upper-division undergraduate or first-year graduate quantum mechanics course. It is also a toolbox for the practicing physicist seeking to automate a variety of algebraic and numerical tasks with the computer. The book is divided into two parts: "Systems in One Dimension" and "Quantum Dynamics." Part I emphasizes topics from a first-year course on quantum mechanics, while Part II includes more advanced topics. Although the text requires some familiarity with Mathematica, appendices are provided for gaining experience with the software and are referenced throughout the book. The text is task-oriented and integrated with numerous problems and exercises, with hints for working on the computer. James M. Feagin is a Professor of Physics at California State University, Fullerton. He was educated at Georgia Tech and the University of North Carolina, Chapel Hill, where he received a Ph.D. in theoretical physics in 1980. He is a Fellow of the Alexander von Humboldt Foundation and has served as visiting Professor at the University of Freiburg, Germany. Feagin is the author of numerous articles on collision physics and the dynamics of few-body systems. He has given a number of invited talks and hosted workshops on incorporating computers into the physics curriculum and is presently helping to introduce computing into the Introductory University Physics Project (IUPP) sponsored by the American Institute of Physics.

Foreword

vii

Preface

ix

Using This Book Interactively

xiii

Part I Systems in One Dimension

3

(234)

Basic Wave Mechanics

5

(8)

Equations of Motion

6

(3)

Stationary States

9

(1)

A Well-Posed Problem

9

(1)

Time-Development Operator

10

(1)

Extra Dimensions

11

(2)

Particle in a Box

13

(18)

Analytical Eigenfunctions

13

(3)

Numerical Eigenfunctions

16

(2)

Two Basic Properties

18

(1)

Rectangular Wave

19

(5)

Quantum Rattle

24

(5)

Measurements

29

(2)

Uncertainty Principle

31

(4)

Free-Particle Wavepacket

35

(10)

Stationary Wavepacket

36

(2)

Moving Wavepacket

38

(7)

Parity

45

(6)

Harmonic Oscillator

51

(24)

Scaled Schrodinger Equation

52

(1)

Method of Solution

53

(1)

Energy Spectrum

54

(4)

Hermite Polynomials

58

(4)

Hypergeometric Functions

62

(5)

Normalized HO Wavefunctions

67

(6)

Raising and Lowering Operators

73

(2)

Variational Method and Perturbation Ideas

75

(10)

HO Ground State Variationally

76

(1)

HO Excited State Variationally

77

(2)

Model Hamiltonian

79

(3)

First-Order Perturbation Energy

82

(3)

Squeezed States

85

(16)

Eigenfunction Expansion

87

(3)

Time Evolution

90

(4)

Newton's Laws

94

(3)

Quasi-Clasical States

97

(4)

Basic Matrix Mechanics

101

(6)

HO Coordinate and Momentum Matrix Elements

102

(1)

HO Coordinate and Momentum Matrices

103

(1)

HO Hamiltonian Matrix

104

(3)

Partial Exact Diagonalization

107

(14)

Model-Hamiltonian Matrix

108

(2)

Matrix Eigenvalues and Eigenvectors

110

(3)

Perturbed Eigenfunctions

113

(2)

Local Energy

115

(1)

Pseudo States and Resonances

116

(1)

Diagonalization

116

(5)

Momentum Representation

121

(24)

Tools

121

(3)

Momentum Wavefunctions

124

(1)

Conventions

125

(1)

HO Momentum Wavefunctions

126

(2)

Dirac Delta Function

128

(3)

Momentum Operator

131

(2)

Local Energy

133

(1)

Coordinate Operator

134

(1)

Momentum-Space Hamiltonian

135

(1)

Exponential Operators

136

(3)

More Squeezed States

139

(6)

Lattice Representation

145

(44)

Coordinate Lattice

145

(5)

Momentum Lattice

150

(3)

Discrete Fourier Transforms

153

(4)

Local Energy

157

(2)

FFT

159

(5)

Wavepacket Propagation

164

(16)

Quantum Diffusion

180

(9)

Morse Oscillator

189

(16)

Kummer's Equation

190

(3)

Eigenenergies

193

(2)

Eigenfunctions

195

(1)

Normalization

196

(4)

Hypergeometric Integrals

200

(5)

Potential Scattering

205

(32)

Numerical Solution

206

(3)

Scattering Amplitudes

209

(6)

Resonance Hunting

215

(3)

Radial Wavefunctions

218

(2)

Resonance Parameterization

220

(5)

Wavepacket Impact

225

(12)

Part II Quantum Dynamics

237

(132)

Quantum Operators

239

(14)

Commutator Algebra

240

(5)

Two-Body Relative Coordinates

245

(4)

Bra-Ket Formalism

249

(2)

Harmonic Oscillator Spectrum

251

(2)

Angular Momentum

253

(14)

Angular Momentum Spectrum

256

(2)

Matrix Representation

258

(3)

New Axis of Quantization

261

(3)

Quantum Rotation Matrix

264

(3)

Angular Momentum Coupling

267

(20)

Spin and Orbital Coupling

271

(3)

Total Angular Momentum Spectrum

274

(3)

Clebsch Gordanary

277

(4)

Wigner 3j Symbols

281

(2)

Recoupling Coefficients

283

(4)

Coordinate and Momentum Representations

287

(16)

Position and Momentum Operators

287

(2)

Commutation Relations

289

(3)

Angular Momentum in Cartesian Coordinates

292

(1)

Rotational Symmetry

293

(3)

Dynamical Symmetry

296

(2)

Runge-Lenz Vector

298

(2)

Hydrogen Atom Spectrum

300

(3)

Angular Momentum in Spherical Coordinates

303

(20)

Spherical Harmonics

308

(5)

New Axis of Quantization

313

(3)

Quantum Rotation Matrix

316

(7)

Hydrogen Atom Schrodinger Equation

323

(22)

Separation in Spherical Coordinates

324

(2)

Bound-State Wavefunctions

326

(7)

Parity

333

(2)

Continuum Wavefunctions

335

(3)

Separation in Parabolic Coordinates

338

(7)

Wavefunctions from the Runge-Lenz Algebra

345

(24)

Raising and Lowering Operators

345

(3)

Top-Rung States

348

(3)

Down the Ladder

351

(2)

Connection with the Parabolic Separation

353

(4)

Linear Stark Effect

357

(7)

Connection with the Spherical Separation

364

(5)

Appendix I. Mathematica Quick view

369

(2)

Appendix II. Notebooks and Basic Tools

371

(12)

AII.1 Projectile Motion Ignoring Air Resistance

371

(7)

AII.2 Including Air Resistance

378

(5)

Appendix III. Home Improvement

383

(40)

AIII.1 Functions

384

(14)

AIII.2 Algebra

398

(17)

AIII.3 Computing

415

(8)

Appendix IV. Quantum Packages

423

(10)

AIV.1 Quantum `Clebsch`

423

(2)

AIV.2 Quantum `integExp`

425

(1)

AIV.3 Quantum `integauss`

425

(1)

AIV.4 Quantum `NonCommutativeMultiply`

426

(1)

AIV.5 Quantum `PowerTools`

427

(1)

AIV.6 Quantum `QuantumRotations`

428

(2)

AIV.7 Quantum `QuickRelm`

430

(1)

AIV.8 Quantum `Trigonometry`

431

(2)

Appendix V. Grad, Div, Curl

433

(32)

AV.1 Vector Products

433

(3)

AV.2 Cartesian Coordinates

436

(9)

AV.3 Curvilinear Coordinates

445

(3)

AV.4 Spherical Coordinates

448

(17)

Bibliography

465

(4)

Index

469

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.