Quantum Mechanics

A comprehensive, mathematically rigorous treatment of non-relativistic quantum mechanics — from the Hilbert-space formalism and the postulates, through the canonical exactly-solvable systems and the standard approximation methods, to the modern structure of the theory (symmetries, scattering, identical particles, entanglement, decoherence) and the measurement problem. The relativistic extension is taken up in the Quantum Field Theory section, to which the final page bridges.

Conventions: $\hslash$ is kept explicit; the inner product is linear in the second argument; state space is a complex separable Hilbert space $\mathcal{H}$, observables are self-adjoint, and closed-system dynamics are unitary.

Contents

Foundations and formalism

1. Mathematical Preliminaries — Hilbert spaces, bra–ket notation, operators, eigenstates, observables, tensor products, density operators.
2. Postulates of Quantum Mechanics — the seven standard postulates: state space, observables, Born rule, collapse, Schrödinger evolution, composite systems, symmetrization.
3. Wave Mechanics and the Position/Momentum Representation — wavefunctions, the canonical commutator, the Schrödinger PDE, probability current, Ehrenfest's theorem, the classical limit.
4. Uncertainty Relations — the Robertson–Schrödinger inequality, $\Delta x\Delta p\ge \hslash /2$, minimum-uncertainty states, the energy–time relation.
5. The Spectral Theorem and Unbounded Operators — self-adjointness, continuous spectra, rigged Hilbert space, Stone's theorem.

Dynamics

6. The Heisenberg Picture — the Schrödinger, Heisenberg, and interaction pictures compared.
7. Unitary Evolution and the Propagator — the evolution operator, time ordering, the propagator, and the bridge to path integrals.
8. Path Integral Formulation — Feynman's sum over paths, Lagrangian recap, Wick rotation.

Exactly-solvable systems

9. The Harmonic Oscillator — ladder operators, the number spectrum, coherent states, the link to Fock space.
10. Piecewise-Constant Potentials and Bound States — wells, barriers, tunneling, the delta potential.
11. Orbital Angular Momentum — the algebra, ladder operators, spherical harmonics, central potentials.
12. Spin and the SU(2) Representation — Stern–Gerlach, Pauli matrices, the $4\pi$ rotation sign.
13. Addition of Angular Momenta — Clebsch–Gordan coefficients, spin–orbit coupling, the Wigner–Eckart theorem.
14. The Hydrogen Atom — the Coulomb spectrum, degeneracy, the hidden $SO(4)$ symmetry, orbitals.
15. Charged Particle in a Magnetic Field: Landau Levels — minimal coupling, Landau levels, gauge choice, the Aharonov–Bohm effect.

Approximation methods

16. Time-Independent Perturbation Theory — the non-degenerate and degenerate series; fine structure, Zeeman, Stark.
17. Time-Dependent Perturbation Theory — the interaction picture, Fermi's golden rule, selection rules, adiabatic/sudden limits.
18. The Variational Method — the variational principle, Rayleigh–Ritz, the helium ground state.
19. The WKB (Semiclassical) Approximation — the $\hslash$-expansion, connection formulae, Bohr–Sommerfeld quantization, tunneling.

Symmetry and scattering

20. Symmetries and Conservation Laws — Wigner's theorem, generators and conserved quantities, degeneracy, the Galilei group.
21. Discrete Symmetries: Parity and Time Reversal — parity selection rules, antiunitary time reversal, Kramers degeneracy.
22. Scattering Theory — cross sections, phase shifts, resonances, the optical theorem, the Born approximation.

Foundations and modern topics

23. Identical Particles and Quantum Statistics — symmetrization, spin–statistics, Pauli exclusion, the exchange force.
24. Density Operators and Open Quantum Systems — mixed states, the partial trace, POVMs, Kraus maps, the Lindblad equation.
25. Entanglement, EPR, and Bell's Theorem — entangled states, the EPR argument, CHSH violation, no-cloning, teleportation.
26. Decoherence and the Classical Limit — environmental entanglement, pointer states, einselection, timescales.
27. The Measurement Problem and Interpretations — the two evolution laws, Copenhagen, many-worlds, Bohmian, objective collapse.

Topics

Focused deep-dives that expand a single programme in full technical and philosophical detail.

• Bohmian Mechanics (the de Broglie–Bohm Pilot-Wave Theory) — the deterministic pilot-wave completion: guidance equation, quantum potential, quantum equilibrium and the Born rule, effective collapse, non-locality; with an in-depth philosophical analysis.

Bridge forward

28. Relativistic Wave Equations: Bridge to Field Theory — Klein–Gordon and Dirac, the negative-energy problem, why fields are needed.

Suggested reading order

• Core course: 1 → 2 → 3 → 4 → 6 → 9 → 10 → 11 → 12 → 14 → 16 → 17 — the backbone of a first graduate QM course.
• Formal foundations: add 5, 7, 8 for the rigorous operator theory, propagator, and path integral.
• Structure and methods: 13, 18, 19, 20, 21, 22 for angular-momentum coupling, non-perturbative methods, symmetry, and scattering.
• Modern foundations: 23 → 24 → 25 → 26 → 27 for identical particles and the quantum-information / measurement-problem cluster.
• Onward: 28 motivates the transition to Quantum Field Theory.