Quantum Mechanics
A Graduate Course

A comprehensive, modern description of Quantum Mechanics at the graduate level, focused on developing the formalism and its applications.

Horatiu Nastase (Author)

9781108838733, Cambridge University Press

Hardback, published 4 August 2022

700 pages
26.1 x 21 x 4 cm, 1.65 kg

Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.

1. The mathematics of Quantum Mechanics 1: Finite dimensional Hilbert spaces
2. The mathematics of Quantum Mechanics 2: Infinite dimensional Hilbert spaces
3. The postulates of Quantum Mechanics and the Schrödinger equation
4. Two-level systems and spin 1/2, entanglement and computation
5. Position and momentum and their bases, canonical quantization, and free particles
6. The Heisenberg uncertainty principle and relations, and Gaussian wave packets
7. One dimensional problems in a potential V (x)
8. The harmonic oscillator 9. The Heisenberg picture and general pictures
evolution operator
10. The Feynman path integral and propagators
11. The classical limit and Hamilton-Jacobi (the WKB method), Ehrenfest theorem
12. Symmetries in quantum mechanics I: Continuous symmetries
13. Symmetries in quantum mechanics II: Discrete symmetries and internal symmetries
14. Theory of angular momentum I: operators, algebras, representations
15. Theory of angular momentum II: addition of angular momenta and representations
oscillator model
16. Applications of angular momentum theory: tensor operators, wave functions and the Schrödinger equation, free particles
17. Spin and L + S
18. The Hydrogen atom
19. General central potential and 3 dimensional (isotropic) harmonic oscillator
20. Systems of identical particles
21. Application of identical particles: He atom (2-electron system) and H2 molecule
22. Quantum mechanics interacting with classical electromagnetism
23. Aharonov-Bohm effect and Berry phase in quantum mechanics
24. Motion in a magnetic field, Hall effect and Landau levels
25. The WKB and semiclassical approximation
26. Bohr-Sommerfeld quantization
27. Dirac quantization condition and magnetic monopole
s 28. Path integrals II: imaginary time and fermionic path integral
29. General theory of quantization of classical mechanics and (Dirac) quantization of constrained systems
30. Quantum entanglement and the EPR paradox
31. The interpretation of quantum mechanics and Bell's inequalities
32. Quantum statistical mechanics and tracing over a subspace
33. Elements of quantum information and quantum computing
34. Quantum complexity and quantum chaos
35. Quantum decoherence and quantum thermalization
36. Time-independent (stationary) perturbation theory: nondegenerate, degenerate, and formal cases
37. Time dependent perturbation theory: first order
38. Time-dependent perturbation theory: second and all orders
39. Application: interaction with (classical) electromagnetic field, absorption, photoelectric and Zeeman effect
40. WKB methods and extensions: state transitions and Euclidean path integrals (instantons)
41. Variational methods
42. Atoms and molecules, orbitals and chemical bonds: Quantum chemistry
43. Nuclear liquid droplet and shell models
44. Interaction of atoms with electromagnetic radiation: transitions and lasers
45. One-dimensional scattering, transfer and S matrices
46. Three dimensional Lippmann-Schwinger equation, scattering amplitudes and cross-sections
47. Born approximation and series, S- and T-matrix
48. Partial wave expansion, phase shift method and scattering length
49. Unitarity, optics and the optical theorem
50. Low energy and bound states, analytical properties of scattering amplitudes
51. Resonances and scattering, complex k and l
52. The semiclassical: WKB and eikonal approximations for scattering
53. Inelastic scattering
54. The Dirac equation
55. Multi-particle states in atoms and condensed matter: Schrödinger vs. occupation number
56. Fock space calculation with field operators
57. The Hartree-Fock approximation and other occupation number approximations
58. Nonstandard statistics: anyons and nonabelions.

Subject Areas: Mathematical physics [PHU], Quantum physics [quantum mechanics & quantum field theory PHQ]