Principles of Quantum Mechanics

1. Mathematical Introduction. - 1. 1. Linear Vector Spaces: Basics. - 1. 2. Inner Product Spaces. - 1. 3. Dual Spaces and the Dirac Notation. - 1. 4. Subspaces. - 1. 5. Linear Operators. - 1. 6. Matrix Elements of Linear Operators. - 1. 7. Active and Passive Transformations. - 1. 8. The Eigenvalue Problem. - 1. 9. Functions of Operators and Related Concepts. - 1. 10. Generalization to Infinite Dimensions. - 2. Review of Classical Mechanics. - 2. 1. The Principle of Least Action and Lagrangian Mechanics. - 2. 2. The Electromagnetic Lagrangian. - 2. 3. The Two-Body Problem. - 2. 4. How Smart Is a Particle? . - 2. 5. The Hamiltonian Formalism. - 2. 6. The Electromagnetic Force in the Hamiltonian Scheme. - 2. 7. Cyclic Coordinates, Poisson Brackets, and Canonical Transformations. - 2. 8. Symmetries and Their Consequences. - 3. All Is Not Well with Classical Mechanics. - 3. 1. Particles and Waves in Classical Physics. - 3. 2. An Experiment with Waves and Particles (Classical). - 3. 3. The Double-Slit Experiment with Light. - 3. 4. Matter Waves (de Broglie Waves). - 3. 5. Conclusions. - 4. The Postulates a General Discussion. - 4. 1. The Postulates. - 4. 2. Discussion of Postulates I -III. - 4. 3. The Schrödinger Equation (Dotting Your i s and Crossing your ? s). - 5. Simple Problems in One Dimension. - 5. 1. The Free Particle. - 5. 2. The Particle in a Box. - 5. 3. The Continuity Equation for Probability. - 5. 4. The Single-Step Potential: a Problem in Scattering. - 5. 5. The Double-Slit Experiment. - 5. 6. Some Theorems. - 6. The Classical Limit. - 7. The Harmonic Oscillator. - 7. 1. Why Study the Harmonic Oscillator? . - 7. 2. Review of the Classical Oscillator. - 7. 3. Quantization of the Oscillator (Coordinate Basis). - 7. 4. The Oscillator in the Energy Basis. - 7. 5. Passage from the Energy Basis to the X Basis. - 8. The Path Integral Formulation of Quantum Theory. - 8. 1. The Path Integral Recipe. - 8. 2. Analysis of the Recipe. - 8. 3. An Approximation to U(t) for the Free Particle. - 8. 4. Path Integral Evaluation of the Free-Particle Propagator. - 8. 5. Equivalence to the Schrödinger Equation. - 8. 6. Potentials of the Form V=a + bx + cx2 + d? + ex? . - 9. The Heisenberg Uncertainty Relations. - 9. 1. Introduction. - 9. 2. Derivation of the Uncertainty Relations. - 9. 3. The Minimum Uncertainty Packet. - 9. 4. Applications of the Uncertainty Principle. - 9. 5. The Energy-Time Uncertainty Relation. - 10. Systems with N Degrees of Freedom. - 10. 1. N Particles in One Dimension. - 10. 2. More Particles in More Dimensions. - 10. 3. Identical Particles. - 11. Symmetries and Their Consequences. - 11. 1. Overview. - 11. 2. Translational Invariance in Quantum Theory. - 11. 3. Time Translational Invariance. - 11. 4. Parity Invariance. - 11. 5. Time-Reversal Symmetry. - 12. Rotational Invariance and Angular Momentum. - 12. 1. Translations in Two Dimensions. - 12. 2. Rotations in Two Dimensions. - 12. 3. The Eigenvalue Problem of Lz. - 12. 4. Angular Momentum in Three Dimensions. - 12. 5. The Eigenvalue Problem of L2 and Lz. - 12. 6. Solution of Rotationally Invariant Problems. - 13. TheHydrogen Atom. - 13. 1. The Eigenvalue Problem. - 13. 2. The Degeneracy of the Hydrogen Spectrum. - 13. 3. Numerical Estimates and Comparison with Experiment. - 13. 4. Multielectron Atoms and the Periodic Table. - 14. Spin. - 14. 1. Introduction. - 14. 2. What is the Nature of Spin? . - 14. 3. Kinematics of Spin. - 14. 4. Spin Dynamics. - 14. 5. Return of Orbital Degrees of Freedom. - 15. Addition of Angular Momenta. - 15. 1. A Simple Example. - 15. 2. The General Problem. - 15. 3. Irreducible Tensor Operators. - 15. 4. Explanation of Some Accidental Degeneracies. - 16. Variational and WKB Methods. - 16. 1. The Variational Method. - 16. 2. The Wentzel-Kramers-Brillouin Method. - 17. Time-Independent Perturbation Theory. - 17. 1. The Formalism. - 17. 2. Some Examples. - 17. 3. Degenerate Perturbation Theory. - 18. Time-Dependent Perturbation Theory. - 18. 1. The Problem. - 18. 2. First-Order Perturbation Theory. - 18. 3. Higher Orders in Perturbation Theory. - 18. 4. A General Discussion of Electromagnetic Interactions. - 18. 5. Interaction of Atoms with Electromagnetic Radiation. - 19. Scattering Theory. - 19. 1. Introduction. - 19. 2. Recapitulation of One-Dimensional Scattering and Overview. - 19. 3. The Born Approximation (Time-Dependent Description). - 19. 4. Born Again (The Time-Independent Approximation). - 19. 5. The Partial Wave Expansion. - 19. 6. Two-Particle Scattering. - 20. The Dirac Equation. - 20. 1. The Free-Particle Dirac Equation. - 20. 2. Electromagnetic Interaction of the Dirac Particle. - 20. 3. More on Relativistic Quantum Mechanics. - 21. Path Integrals II. - 21. 1. Derivation of the Path Integral. - 21. 2. Imaginary Time Formalism. - 21. 3. Spin and Fermion Path Integrals. - 21. 4. Summary. - A. l. Matrix Inversion. - A. 2. Gaussian Integrals. - A. 3. Complex Numbers.