Principles of Quantum Mechanics

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

Wilkin Collin praised the book for early introduction of Dirac's notation.^[5] Gino Segrè called it an "excellent text" that elucidates the mathematical foundations and postulates of quantum mechanics in a pedagogical manner.^[6]

• Modern Quantum Mechanics by J. J. Sakurai
• List of textbooks on classical and quantum mechanics