Quantum Principles and Particles

601 Pages·2018·14.406 MB·English

by  Walter Wilcox

Quantum Principles & Particles
Textbook Series in Physical Sciences
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Quantum Principles and Particles, Second Edition
Walter Wilcox
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Quantum Principles & Particles
Walter M. Wilcox
CRC Press
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Contents
Preface to the Second Edition .........................................................................................................................................................xiii
Preface to the First Edition ..................................................................................................................................................................xv
Section i—Quantum Principles
chapter 1 Perspective and Principles .....................................................................................................................................3
1.1 Prelude to Quantum Mechanics .........................................................................................................3
1.2 Stern–Gerlach Experiment .....................................................................................................................7
1.3 Idealized Stern–Gerlach Results ........................................................................................................12
1.4 Classical Model Attempts ......................................................................................................................15
1.5 Wave Functions for Two-Physical-Outcomes Cases .............................................................20
1.6 Measurement Symbols and Completeness ...............................................................................24
1.7 Process Diagrams and Operator Properties ...............................................................................26
1.8 Operator Reformulation .........................................................................................................................36
1.9 Operator Rotation ......................................................................................................................................38
1.10 Bra–Ket Notation/Basis States ...........................................................................................................44
1.11 Transition Amplitudes ............................................................................................................................46
1.12 Three-Magnet Setup Example—Coherence ............................................................................49
1.13 Hermitian Conjugation ...........................................................................................................................51
1.14 Unitary Operators ......................................................................................................................................55
1.15 A Very Special Operator .........................................................................................................................56
1.16 Matrix Representations ...........................................................................................................................57
1.17 Matrix Wave Function Recovery .......................................................................................................61
1.18 Expectation Values ....................................................................................................................................63
1.19 Wrap-Up .........................................................................................................................................................64
Problems ..........................................................................................................................................................65
chapter 2 Particle Motion in One Dimension .................................................................................................................75
2.1 Photoelectric Effect ..................................................................................................................................75
2.2 Compton Effect ...........................................................................................................................................77
2.3 Uncertainty Relation for Photons .....................................................................................................81
2.4 Stability of Ground States ......................................................................................................................82
2.5 Bohr Model ...................................................................................................................................................84
2.6 Fourier Transform and Uncertainty Relations ...........................................................................85
2.7 Schrödinger Equation..............................................................................................................................88
2.8 Schrödinger Equation Example .........................................................................................................92
2.9 Dirac Delta Functions ..............................................................................................................................95
2.10 Wave Functions and Probability .......................................................................................................97
vii
viii Contents
2.11 Probability Current .................................................................................................................................100
2.12 Time Separable Solutions ..................................................................................................................101
2.13 Completeness for Particle States ...................................................................................................102
2.14 Particle Operator Properties .............................................................................................................105
2.15 Operator Rules ..........................................................................................................................................108
2.16 Time Evolution and Expectation Values ....................................................................................111
2.17 Wrap-Up ........................................................................................................................................................112
Problems ........................................................................................................................................................113
chapter 3 Some One-Dimensional Solutions to the Schrödinger Equation ..............................................119
3.1 Introduction ................................................................................................................................................119
3.2 The Infinite Square Well: Differential Solution ........................................................................121
3.3 The Infinite Square Well: Operator Solution ...........................................................................127
3.4 The Finite Potential Barrier Step Potential ................................................................................132
3.5 The Harmonic Oscillator .....................................................................................................................137
3.6 The Attractive Kronig–Penney Model ........................................................................................146
3.7 Bound State and Scattering Solutions .......................................................................................153
Problems .......................................................................................................................................................154
chapter 4 Hilbert Space and Unitary Transformations............................................................................................159
4.1 Introduction and Notation ................................................................................................................159
4.2 Inner and Outer Operator Products .............................................................................................161
4.3 Operator–Matrix Relationship .........................................................................................................163
4.4 Hermitian Operators and Eigenkets ............................................................................................164
4.5 Gram–Schmidt Orthogonalization Process ............................................................................166
4.6 Compatible Operators .........................................................................................................................167
Theorem .......................................................................................................................................................167
4.7 Uncertainty Relations and Incompatible Operators ..........................................................169
4.8 Simultaneously Measureable Operators ...................................................................................172
4.9 Unitary Transformations and Change of Basis .......................................................................173
4.10 Coordinate Displacements and Unitary Transformations ..............................................177
4.11 Schrödinger and Heisenburg Pictures of Time Evolution ..............................................179
4.12 Free Gaussian Wave Packet in the Heisenberg Picture ....................................................181
4.13 Potentials and the Ehrenfest Theorem ......................................................................................182
Problems .......................................................................................................................................................184
chapter 5 Three Static Approximation Methods ........................................................................................................189
5.1 Introduction ...............................................................................................................................................189
5.2 Time-Independent Perturbation Theory .................................................................................190
5.3 Examples of Time-Independent Perturbation Theory ....................................................193
5.4 Aspects of Degenerate Perturbation Theory ........................................................................195
5.5 WKB Semiclassical Approximation ...............................................................................................196
5.6 Use of the WKB Approximation in Barrier Penetration.....................................................200
5.7 Use of the WKB Approximation in Bound States .................................................................201
5.8 Variational Methods ...............................................................................................................................205
Problems .......................................................................................................................................................209
Contents ix
chapter 6 Generalization to Three Dimensions ..........................................................................................................215
6.1 Cartesian Basis States and Wave Functions in Three Dimensions .............................215
6.2 Position/Momentum Eigenket Generalization ......................................................................217
6.3 Example: Three-Dimensional Infinite Square Well ..............................................................219
6.4 Spherical Basis States ............................................................................................................................221
6.5 Orbital Angular Momentum Operator .......................................................................................222
6.6 Effect of Angular Momentum on Basis States .......................................................................223
6.7 Energy Eigenvalue Equation and Angular Momentum ..................................................227
6.8 Complete Set of Observables for the Radial Schrödinger Equation ........................229
6.9 Specification of Angular Momentum Eigenstates ..............................................................231
6.10 Angular Momentum Eigenvectors and Spherical Harmonics .....................................237
6.11 Completeness and Other Properties of Spherical Harmonics ....................................241
6.12 Radial Eigenfunctions ...........................................................................................................................244
Problems .......................................................................................................................................................245
Section ii—Quantum Particles
chapter 7 The Three-Dimensional Radial Equation ..................................................................................................251
7.1 Recap of the Situation ..........................................................................................................................251
7.2 The Free Particle ......................................................................................................................................253
7.3 The Infinite Spherical Well Potential ............................................................................................258
7.4 The “Deuteron” .........................................................................................................................................262
7.5 The Coulomb Potential: Initial Considerations ......................................................................274
7.6 The Coulomb Potential: 2-D Harmonic Oscillator Comparison ..................................277
7.7 The Confined Coulombic Model ...................................................................................................292
Problems .......................................................................................................................................................299
chapter 8 Addition of Angular Momenta .......................................................................................................................309
8.1 General Angular-Momentum Eigenstate Properties.........................................................309
8.2 Combining Angular Momenta for Two Systems ...................................................................311
8.3 Explicit Example of Adding Two Spin-1/2 Systems .............................................................315
8.4 Explicit Example of Adding Orbital Angular Momentum and Spin 1/2 .................317
8.5 Hydrogen Atom and the Choice of Basis States ...................................................................322
8.6 Hydrogen Atom and Perturbative Energy Shifts .................................................................325
Problems .......................................................................................................................................................329
chapter 9 Spin and Statistics ...................................................................................................................................................335
9.1 The Connection between Spin and Statistics .......................................................................335
9.2 Building Wave Functions with Identical Particles ................................................................336
9.3 Particle Occupation Basis ...................................................................................................................339
9.4 More on Fermi–Dirac Statistics .......................................................................................................343
9.5 Interaction Operator and Feynman Diagrams ......................................................................346
9.6 Implications of Detailed Balance ...................................................................................................348
9.7 Density of States Expressions ..........................................................................................................353