Lectures on Quantum Mechanics: A Primer for Mathematicians
Cambridge University Press, 9/17/2020
EAN 9781108429764, ISBN10: 1108429769
Hardcover, 582 pages, 23.4 x 15.5 x 3.3 cm
Language: English
Originally published in English
Quantum mechanics is one of the principle pillars of modern physics. It also remains a topic of great interest to mathematicians. Since its discovery it has inspired and been inspired by many topics within modern mathematics, including functional analysis and operator algebras, Lie groups, Lie algebras and their representations, principle bundles, distribution theory, and much more. Written with beginning graduate students in mathematics in mind, this book provides a thorough treatment of (nonrelativistic) quantum mechanics in a style that is leisurely, without the usual theorem-proof grammar of pure mathematics, while remaining mathematically honest. The author takes the time to fully develop the required mathematics and employs a consistent mathematical presentation to clarify the often-confusing notation of physics texts. Along the way the reader encounters several topics requiring more advanced mathematics than found in many discussions of the subject, making for a fascinating course in how mathematics and physics interact.
Preface
Prolegomenon
1. The Harmonic Oscillator
Classical verses Quantum
2. The Mathematical Structure of Quantum Mechanics
3. Observables and Expectation Values
4. The Projection Postulate Examined
5. Rigged Hilbert Space and the Dirac Calculus
6. A Review of Classical Mechanics
7. Hamilton-Jacobi Theory *
8. Classical Mechanics Regain'd
9. Wave Mechanics I
Heisenberg Uncertainty
10. Wave Mechanics II
The Fourier Transform
11. Wave Mechanics III
The Quantum Oscillator
12. Angular Momentum I
Basics
13. Angular Momentum II
Representations of su(2)
14. Angular Momentum III
The Central Force Problem
15. Wave Mechanics IV
The Hydrogenic Potential
16. Wave Mechanics V
Hidden Symmetry Revealed
17. Wave Mechanics VI
Hidden Symmetry Solved
18. Angular Momentum IV
Addition Rules and Spin
19. Wave Mechanics VII
Pauli's Spinor Theory
20. Clifford Algebras and Spin Representations *
21. Many-Particle Quantum Systems
22. The EPR Argument and Bell's Inequalities
23. Ensembles and Density Operators
24. Bosons and Fermions
25. The Fock Space for Indistinguishable Quanta
26. An Introduction to Quantum Statistical Mechanics
27. Quantum Dynamics
28. Unitary Representations and Conservation Laws
29. The Feynman Formulation of Quantum Mechanics
30. A Mathematical Interlude
Gaussian Integrals
31. Evaluating Path Integrals I
32. Evaluating Path Integrals II
Epilogue
Resources for Individual Exploration
Bibliography
Index.