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PART Ⅰ ELEMENTARY PRINCIPLES AND APPLICATIONS TO PROBLEMS IN ONE DIMENSION 1

1 Review of Concepts of Classical Mechanics 3

1.1 Generalized or “Good” Coordinates 3

1.2 Energy，the Hamiltonian，and Angular Momentum 6

1.3 The State of a System 19

1.4 Properties of the One-Dimensional Potential Function 24

2 Historical Review: Experiments and Theories 30

2.1 Dates 30

2.2 The Work of Planck. Blackbody Radiation 31

2.3 The Work of Einstein. The Photoelectric Effect 36

2.4 The Work of Bohr. A Quantum Theory of Atomic States 39

2.5 Waves versus Particles 43

2.6 The de Broglie Hypothesis and the Davisson-Germer Experiment 46

2.7 The Work of Heisenberg. Uncertaintu as a Cornerstone of Natural Law 53

2.8 The Work of Born. Probabiliry Waves 55

2.9 Semiphilosophical Epilogue to Chapter 2 57

3 The Postulates of Quantum Mechanics. Operators，Eigenfunctions，and Eigenvalues 68

3.1 Observables and Operators 68

3.2 Measurement in Quantum Mechanics 74

3.3 The State Function and Expectation Values 76

3.4 Time Development of the State Function 80

3.5 Solution to the Initial-Value Problem in Quantum Mechanics 84

4 Preparatory Concepts. Function Spaces and Hermitian Operators 90

4.1 Particle in a Box and Further Remarks on Normalization 90

4.2 The Bohr Correspondence Principle 94

4.3 Dirac Notation 97

4.4 Hilbert Space 98

4.5 Hermitian Operators 104

4.6 Properties of Hermitian Operators 108

5 Superposition and Compatible Observables 115

5.1 The Superposition Principle 115

5.2 Commutator Relations in Quantum Mechanics 130

5.3 More on the Commutator Theorem 137

5.4 Commutator Relations and the Uncertainty Principle 140

5.5 “Complete” Sets of Commuting Observables 143

6 Time Development，Conservation Theorems，and Parity 152

6.1 Time Development of State Functions 152

6.2 Time Development of Expectation Values 168

6.3 Conservation of Energy，Linear and Angular Momentum 171

6.4 Conservation of Parity 176

7 Additional One-Dimensional Problems. Bound and Unbound States 187

7.1 General Properties of the One-Dimensional Schrodinger Equation 187

7.2 The Harmonic Oscillator 190

7.3 Eigenfunctions of the Harmonic Oscillator Hamiltonian 198

7.4 The Harmonic Oscillator in Momentum Space 211

7.5 Unbound States 216

7.6 One-Dimensional Barrier Problems 222

7.7 The Rectangular Barrier. Tunneling 228

7.8 The Ramsauer Effect 235

7.9 Kinetic Properties of a Wave Packet Scattered from a Potential Barrier 241

7.10 The WKB Approximation 243

7.11 Principle of Least Action and Feyntnan’s Path Integral Formulation 268

8 Finite Potential Well，Periodic Lattice，and Some Simple Problems with Two Degrees of Freedom 278

8.1 The Finite Potential Well 278

8.2 Periodic Lattice. Energy Gaps 289

8.3 Standing Waves at the Band Edges 307

8.4 Brief Qualitative Description of the Theory of Conduction in Solids 313

8.5 Two Beads on a Wire and a Particle in a Two-Dimensional Box 317

8.6 Two-Dimensional Harmonic Oscillator 324

8.7 Linear Combination of Atomic Orbitals （LCAO） Approximation 331

8.8 Density of States in Various Dimensions 336

PART Ⅱ FURTHER DEVELOPMENT OF THE THEORY AND APPLICATIONS TO PROBLEMS IN THREE DIMENSIONS 347

9 Angular Momentum 349

9.1 Basic Properties 349

9.2 Eigenvalues of the Angular Momentum Operators 358

9.3 Eigenfunctions of the Orbital Angular Momentum Operators L2 and Lz 367

9.4 Addition of Angular Momentum 386

9.5 Total Angular Momentum for Two or More Electrons 396

10 Problems in Three Dimensions 404

10.1 The Free Particle in Cartesian Coordinates 404

10.2 The Free Particle in Spherical Coordinates 410

10.3 The Free-Particle Radial Wavefunction 415

10.4 A Charged Particle in a Magnetic Field 430

10.5 The Two-Particle Problem 436

10.6 The Hydrogen Atom 446

10.7 Elementary Theory of Radiation 463

10.8 Thomas-Fermi Model 472

11 Elements of Matrix Mechanics. Spin Wavefunctions 480

11.1 Basis and Representations 481

11.2 Elementary Matrix Properties 488

11.3 Unitary and Similariry Transformations in Quantum Mechanics 492

11.4 The Energy Representation 499

11.5 Angular Momentum Matrices 504

11.6 The Pauli Spin Matrices 512

11.7 Free-Particle Wavefunctions，Including Spin 517

11.8 The Magnetic Moment of an Electron 519

11.9 Precession of an Electron in a Magnetic Field 527

11.10 The Addition of Two Spins 536

11.11 The Density Matrix 543

11.12 Other “Pictures” in Quantum Mechanics 553

11.13 Polarization States. EPR Revisited 558

11.14 The Transfer Matrix 571

12 Application to Atomic，Molecular，Solid-State，and Nuclear Physics. Elements of Quantum Statistics 579

12.1 The Total Angular Momentum，J 579

12.2 One-Electron Atoms 584

12.3 The Pauli Principle 597

12.4 The Periodic Table 602

12.5 The Slater Determinant 612

12.6 Application of Symmetrization Rules to the Helium Atom 614

12.7 The Hydrogen and Deuterium Molecules 623

12.8 Brief Description of Quantum Models for Superconductivity and Superfluidity 630

12.9 Impurity Semiconductors and the p-n Junction 641

12.10 Elements of Nuclear Physics. The Deuteron and Isospin 669

13 Perturbation Theory 681

13.1 Time-Independent，Nondegenerate Perturbation Theory 681

13.2 Time-Independent，Degenerate Perturbation Theory 692

13.3 The Stark Effect 700

13.4 The Nearly Free Electron Model 703

13.5 Time-Dependent Perturbation Theory 709

13.6 Harmonic Perturbation 712

13.7 Application of Harmonic Perturbation Theory 718

13.8 Selective Perturbations in Time 727

13.9 Atom-Radiation Interaction 739

13.10 Hartree-Fock Model 757

14 Scattering in Three Dimensions 762

14.1 Partial Waves 762

14.2 S-Wave Scattering 770

14.3 Center-of-Mass Frame 774

14.4 The Born Approximation 777

14.5 Atomic-Radiative Absorption Cross Section 782

14.6 Elements of Formal Scattering Theory. The Lippmatm-Schwinger Equation 785

15 Relativistic Quantum Mechanics 793

15.1 Preliminary Remarks 793

15.2 Klein-Gordon Equation 798

15.3 Dirac Equation 800

15.4 Electron Magnetic Moment 806

15.5 Covariant Description 810

16 Quantum Computing 817

16.1 Binary Number System 817

16.2 Logic Gates 823

16.3 Turing Machine and Complexity Classes 830

16.4 Qubits and Quantum Logic Gates 832

List of Symbols 843

APPENDIXES 847

A Additional Remarks on the x and p Representations 849

B Spin and Statistics 853

C Representations of the Delta Function 857

D Differential Vector Relations 861

E Harmonic Oscillator in Spherical Coordinates 865

F Physical Constants and Equivalence Relations 867

Index 871