Lectures on Quantum Mechanics for Mathematics Students

L. D. Faddeev and O. A. Yakubovskii

Published by Universities Press, 2020

About this Item

Contents: Preface. 1. Preface to the English edition. 2. The algebra of observables in classical mechanics. 3. States. 4. Liouville s theorem, and two pictures of motion in classical mechanics. 5. Physical bases of quantum mechanics. 6. A finite-dimensional model of quantum mechanics. 7. States in quantum mechanics. 8. Heisenberg uncertainty relations. 9. Physical meaning of the eigenvalues and eigenvectors of observables. 10. Two pictures of motion in quantum mechanics. The Schrödinger equation. Stationary states. 11. Quantum mechanics of real systems. The Heisenberg commutation relations. 12. Coordinate and momentum representations. 13. Eigenfunctions of the operators Q and P. 14. The energy, the angular momentum, and other examples of observables. 15. The interconnection between quantum and classical mechanics. Passage to the limit from quantum mechanics to classical mechanics. 16. One-dimensional problems of quantum mechanics. A free one-dimensional particle. 17. The harmonic oscillator. 18. The problem of the oscillator in the coordinate representation. 19. Representation of the states of a one-dimensional particle in the sequence space l2. 20. Representation of the states for a one-dimensional particle in the space D of entire analytic functions. 21. The general case of one-dimensional motion. 22. Three-dimensional problems in quantum mechanics. A three-dimensional free particle. 23. A three-dimensional particle in a potential field. 24. Angular momentum. 25. The rotation group. 26. Representations of the rotation group. 27. Spherically symmetric operators. 28. Representation of rotations by 2×2 unitary matrices. 29. Representation of the rotation group on a space of entire analytic functions of two complex variables. 30. Uniqueness of the representations Dj. 31. Representations of the rotation group on the space L2(S2) Spherical functions. 32. The radial Schrödinger equation. 33. The hydrogen atom. The alkali metal atoms. 34. Perturbation theory. 35. The variational principle. 36. Scattering theory. Physical formulation of the problem. 37. Scattering of a one-dimensional particle by a potential barrier. 38. Physical meaning of the solutions ? 1 and ? 2. 39. Scattering by a rectangular barrier.40. Scattering by a potential center. 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. While 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.

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Bibliographic Details

• Title: Lectures on Quantum Mechanics for Mathematics Students
• Publisher: Universities Press
• Publication Date: 2020
• Binding: Soft cover
• Condition: New