Lectures on Quantum Mechanics for Mathematics Students

✍ Scribed by L. D. Faddeev, Oleg Aleksandrovich Yakubovskiĭ

• Publisher: American Mathematical Society
• Year: 2009
• Tongue: English
• Leaves: 248
• Series: Student Mathematical Library 47
• Category: Library

✦ Synopsis

This book is based on notes from the course developed and taught for more than 30 years at the Department of Mathematics of Leningrad University. The goal of the course was to present the basics of quantum mechanics and its mathematical content to students in mathematics. This book differs from the majority of other textbooks on the subject in that much more attention is paid to general principles of quantum mechanics. In particular, the authors describe in detail the relation between classical and quantum mechanics. When selecting particular topics, the authors emphasize those that are related to interesting mathematical theories. In particular, the book contains a discussion of problems related to group representation theory and to scattering theory. This book is rather elementary and concise, and it does not require prerequisites beyond the standard undergraduate mathematical curriculum. It is aimed at giving a mathematically oriented student the opportunity to grasp the main points of quantum theory in a mathematical framework.

✦ Table of Contents

Cover......Page 1
Title: Lectures on QuantumMechanics forMathematics Students......Page 3
ISBN 978-0-8218-4699-5......Page 4
Contents......Page 6
Preface......Page 10
Preface to the English Edition......Page 12
§ 1. The algebra of observables in classicalmechanics......Page 14
§ 2. States......Page 19
§ 3. Liouville's theorem, and two pictures of motion in classical mechanics......Page 26
§ 4. Physical bases of quantum mechanics......Page 28
5. A finite-dimensional model of quantum mechanics......Page 40
§ 6. States in quantum mechanics......Page 45
§ 7. Heisenberg uncertainty relations......Page 49
§ 8. Physical meaning of the eigenvalues and eigenvectors of observables......Page 52
§ 9. Two pictures of motion in quantum mechanics. The Schrodinger equation. Stationary states......Page 57
§ 10. Quantum mechanics of real systems. The Heisenberg commutation relations......Page 62
§ 11. Coordinate and momentum representations......Page 67
§ 12. "Eigenfunctions" of the operators Q and P......Page 73
§ 13. The energy, the angular momentum, and other examples of observables......Page 76
§ 14. The interconnection between quantum and classical mechanics. Passage to the limit from quantum mechanics to classical mechanics......Page 82
§ 15. One-dimensional problems of quantum mechanics. A free one-dimensional particle......Page 90
§ 16. The harmonic oscillator......Page 96
§ 17. The problem of the oscillator in the coordinate representation......Page 100
§ 18. Representation of the states of a one-dimensional particle in the sequence space 12......Page 103
§ 19. Representation of the states for a one-dimensional particle in the space D of entire analytic functions......Page 107
§ 20. The general case of one-dimensional motion......Page 108
§ 21. Three-dimensional problems in quantum mechanics. A three-dimensional free particle......Page 116
§ 22. A three-dimensional particle in a potential field......Page 117
§ 23. Angular momentum......Page 119
§ 24. The rotation group......Page 121
§ 25. Representations of the rotation group......Page 124
§ 26. Spherically symmetric operators......Page 127
§ 27. Representation of rotations by 2 x 2 unitary matrices......Page 130
§ 28. Representation of the rotation group on a space of entire analytic functions of two complex variables......Page 133
§ 29. Uniqueness of the representations Dj......Page 136
§ 30. Representations of the rotation group on the space L2(S2). Spherical functions......Page 140
§ 31. The radial Schrodinger equation......Page 143
§ 32. The hydrogen atom. The alkali metal atoms......Page 149
§ 33. Perturbation theory......Page 160
§ 34. The variational principle......Page 167
§ 35. Scattering theory. Physical formulation of the problem......Page 170
§ 36. Scattering of a one-dimensional particle by a potential barrier......Page 172
§ 37. Physical meaning of the solutions $\psi _1$ and $\psi _2$......Page 177
§ 38. Scattering by a rectangular barrier......Page 180
§ 39. Scattering by a potential center......Page 182
§ 40. Motion of wave packets in a central force field......Page 188
41. The integral equation of scattering theory......Page 194
§ 42. Derivation of a formula for the cross-section......Page 196
§ 43. Abstract scattering theory......Page 201
§ 44. Properties of commuting operators......Page 210
§ 45. Representation of the state space with respect to a complete set of observables......Page 214
§ 46. Spin......Page 216
§ 47. Spin of a system of two electrons......Page 221
§ 48. Systems of many particles. The identity principle......Page 225
49. Symmetry of the coordinate wave functions of a system of two electrons. The helium atom......Page 228
§ 50. Multi-electron atoms. One-electron approximation......Page 230
§ 51. The self-consistent field equations......Page 236
§ 52. Mendeleev's periodic system of the elements......Page 239
Appendix: Lagrangian Formulation of Classical Mechanics......Page 244
Titles in This Series......Page 248

✦ Subjects

Физика;Квантовая физика;