Principles of Quantum Mechanics – R. Shankar – 1st Edition

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. Translational Invariance in Quantum Theory
11.2. Time Translational Invariance
11.3. Parity Invariance
11.4. 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 L
12.6. Solution of Rotationally Invariant Problems
13. The Hydrogen 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
Appendix
A.1. Matrix Inversion
A.2. Gaussian Integrals
A.3. Complex Numbers
A.4. The i? Prescription
ANSWERS TO SELECTED EXERCISES