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1.Mathematical Introduction1

1.1.Linear Vector Spaces： Basics1

1.2.Inner Product Spaces7

1.3.The Dirac Notation13

1.4.Subspaces20

1.5.Linear Operators21

1.6.Matrix Elements of Linear Operators23

1.7.Active and Passive Transformations32

1.8.The Eigenvalue Problem33

1.9.Functions of Operators and Related Concepts58

1.10.Generalization to Infinite Dimensions61

2.Review of Classical Mechanics79

2.1.The Principle of Least Action and Lagrangian Mechanics79

2.2.The Electromagnetic Lagrangian88

2.3.The Two-Body Problem89

2.4.How Smart Is a Particle？91

2.5.The Hamiltonian Formalism91

2.6.The Electromagnetic Force in the Hamiltonian Scheme95

2.7.Cyclic Coordinates， Poisson Brackets， and Canonical Transformations96

2.8.Symmetries and Their Consequences104

3.All Is Not Well with Classical Mechanics111

3.1.Particles and Waves in Classical Physics111

3.2.An Experiment with Waves and Particles （Classical）112

3.3.The Double-Slit Experiment with Light115

3.4.Matter Waves （de Broglie Waves）117

3.5.Conclusions118

4.The Postulates—A General Discussion119

4.1.The Postulates119

4.2.Discussion of Postulates Ⅰ-Ⅲ121

4.3.The Schrodinger Equation151

5.Simple Problems in One Dimension159

5.1.The Free Particle159

5.2.The Particle in a Box165

5.3.The Continuity Equation for Probability174

5.4.The Single-Step Potential：A Problem in Scattering177

5.5.The Double-Slit Experiment186

5.6.Some Theorems186

6.The Classical Limit189

7.The Harmonic Oscillator195

7.1.Why Study the Harmonic Oscillator？195

7.2.Review of the Classical Oscillator199

7.3.Quantization of the Oscillator （Coordinate Basis）199

7.4.The Oscillator in the Energy Basis214

7.5.Passage from the Energy Basis to the X Basis227

8.The Path Integral Formulation of Quantum Theory233

8.1.The Path Integral Recipe233

8.2.Analysis of the Recipe234

8.3.An Approximation to U（t） for the Free Particle235

8.4.Path Integral Evaluation of the Free-Particle Propagator237

8.5.Equivalence to the Schrodinger Equation240

8.6.Potentials of the form V=a+bx+cx2+dx+exx242

9.The Heisenberg Uncertainty Relations245

9.1.Introduction245

9.2.Derivation of the Uncertainty Relations245

9.3.The Minimum Uncertainty Packet247

9.4.Applications of the Uncertainty Principle249

9.5.The Energy-Time Uncertainty Relation252

10.Systems with N Degrees of Freedom255

10.1.N Particles in One Dimension255

10.2.More Particles in More Dimensions267

10.3.Identical Particles268

11.Symmetries and Their Consequences289

11.1.Overview289

11.2.Translational Invariance in Quantum Theory289

11.3.Time Trans lational Invariance305

11.4.Parity Invariance307

12.Rotational Invariance and Angular Momentum313

12.1.Translations in Two Dimensions313

12.2.Rotations in Two Dimensions314

12.3.The Eigenvalue Problem of Lz321

12.4.Angular Momentum in Three Dimensions326

12.5.The Eigenvalue Problem of L2 and Lz330

12.6.Solution of Rotationally Invariant Problems349

13.The Hydrogen Atom363

13.1.The Eigenvalue Problem363

13.2.The Degeneracy of the Hydrogen Spectrum369

13.3.Numerical Estimates and Comparison with Experiment371

13.4.Multielectron Atoms and the Periodic Table379

14.Spin383

14.1.Introduction383

14.2.What is the Nature of Spin？383

14.3.Kinematics of Spin384

14.4.Spin Dynamics396

14.5.Return of Orbital Degrees of Freedom408

15.Addition of Angular Momenta413

15.1.A Simple Example413

15.2.The General Problem418

15.3.Irreducible Tensor Operators426

15.4.Explanation of Some “Accidental” Degeneracies432

16.Variational and WKB Methods439

16.1.The Variational Method439

16.2.The Wentzel-Kramers-Brillouin Method446

17.Time-Independent Perturbation Theory459

17.1.The Formalism459

17.2.Some Examples462

17.3.Degenerate Perturbation Theory473

18.Time-Dependent Perturbation Theory481

18.1.The Problem481

18.2.First-Order Perturbation Theory482

18.3.Higher Orders in Perturbation Theory492

18.4.A General Discussion of the Electromagnetic Interactions501

18.5.Interaction of Atoms with Electromagnetic Radiation508

19.Scattering Theory533

19.1.Introduction533

19.2.Recapitulation of One-Dimensional Scattering and Overview534

19.3.The Born Approximation （Time-Dependent Description）540

19.4.Born Again （Time-Independent Approximation）545

19.5.The Partial Wave Expansion556

19.6.Two-Particle Scattering567

20.The Dirac Equation575

20.1.The Free-Particle Dirac Equation575

20.2.Electromagnetic Interaction of the Dirac Particle578

20.3.More on Relativistic Quantum Mechanics586

Appendix593

A.1.Matrix Inversion593

A.2.Gaussian Integrals596

A.3.Complex Numbers598

ANSWERS TO SELECTED EXERCISES601

TABLE OF CONSTANTS605

INDEX607