Softcover ISBN:  9780821846995 
Product Code:  STML/47 
List Price:  $44.00 
Individual Price:  $35.20 
Electronic ISBN:  9781470416331 
Product Code:  STML/47.E 
List Price:  $41.00 
Individual Price:  $32.80 

Book DetailsStudent Mathematical LibraryVolume: 47; 2009; 234 ppMSC: Primary 81;
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.ReadershipUndergraduate and graduate students interested in learning the basics of quantum mechanics.

Table of Contents

Chapters

1. The algebra of observables in classical mechanics

2. States

3. Liouville’s theorem, and two pictures of motion in classical mechanics

4. Physical bases of quantum mechanics

5. A finitedimensional model of quantum mechanics

6. States in quantum mechanics

7. Heisenberg uncertainty relations

8. Physical meaning of the eigenvalues and eigenvectors of observables

9. Two pictures of motion in quantum mechanics. The Schrödinger equation. Stationary states

10. Quantum mechanics of real systems. The Heisenberg commutation relations

11. Coordinate and momentum representations

12. “Eigenfunctions” of the operators $Q$ and $P$

13. The energy, the angular momentum, and other examples of observables

14. The interconnection between quantum and classical mechanics. Passage to the limit from quantum mechanics to classical mechanics

15. Onedimensional problems of quantum mechanics. A free onedimensional particle

16. The harmonic oscillator

17. The problem of the oscillator in the coordinate representation

18. Representation of the states of a onedimensional particle in the sequence space $l_2$

19. Representation of the states for a onedimensional particle in the space $\mathcal {D}$ of entire analytic functions

20. The general case of onedimensional motion

21. Threedimensional problems in quantum mechanics. A threedimensional free particle

22. A threedimensional particle in a potential field

23. Angular momentum

24. The rotation group

25. Representations of the rotation group

26. Spherically symmetric operators

27. Representation of rotations by $2\times 2$ unitary matrices

28. Representation of the rotation group on a space of entire analytic functions of two complex variables

29. Uniqueness of the representations $D_j$

30. Representations of the rotation group on the space $L^2(S^2)$. Spherical functions

31. The radial Schrödinger equation

32. The hydrogen atom. The alkali metal atoms

33. Perturbation theory

34. The variational principle

35. Scattering theory. Physical formulation of the problem

36. Scattering of a onedimensional particle by a potential barrier

37. Physical meaning of the solutions $\psi _1$ and $\psi _2$

38. Scattering by a rectangular barrier

39. Scattering by a potential center

40. Motion of wave packets in a central force field

41. The integral equation of scattering theory

42. Derivation of a formula for the crosssection

43. Abstract scattering theory

44. Properties of commuting operators

45. Representation of the state space with respect to a complete set of observables

46. Spin

47. Spin of a system of two electrons

48. Systems of many particles. The identity principle

49. Symmetry of the coordinate wave functions of a system of two electrons. The helium atom

50. Multielectron atoms. Oneelectron approximation

51. The selfconsistent field equations

52. Mendeleev’s periodic system of the elements

53. Appendix. Lagrangian formulation of classical mechanics


Additional Material

Reviews

The present volume has several desirable features. It speaks to mathematicians broadly, not merely practitioners of some narrow specialty. It faithfully explains physical ideas/concerns, rather than addresses the mathematician eager only to glean from physics a purely mathematical problem to attack. This book accomplishes its task as quickly as one could hope but still achieves interesting applications...Highly recommended.
D.V. Feldman, Choice


RequestsReview Copy – for reviewers who would like to review an AMS bookPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
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.
Undergraduate and graduate students interested in learning the basics of quantum mechanics.

Chapters

1. The algebra of observables in classical mechanics

2. States

3. Liouville’s theorem, and two pictures of motion in classical mechanics

4. Physical bases of quantum mechanics

5. A finitedimensional model of quantum mechanics

6. States in quantum mechanics

7. Heisenberg uncertainty relations

8. Physical meaning of the eigenvalues and eigenvectors of observables

9. Two pictures of motion in quantum mechanics. The Schrödinger equation. Stationary states

10. Quantum mechanics of real systems. The Heisenberg commutation relations

11. Coordinate and momentum representations

12. “Eigenfunctions” of the operators $Q$ and $P$

13. The energy, the angular momentum, and other examples of observables

14. The interconnection between quantum and classical mechanics. Passage to the limit from quantum mechanics to classical mechanics

15. Onedimensional problems of quantum mechanics. A free onedimensional particle

16. The harmonic oscillator

17. The problem of the oscillator in the coordinate representation

18. Representation of the states of a onedimensional particle in the sequence space $l_2$

19. Representation of the states for a onedimensional particle in the space $\mathcal {D}$ of entire analytic functions

20. The general case of onedimensional motion

21. Threedimensional problems in quantum mechanics. A threedimensional free particle

22. A threedimensional particle in a potential field

23. Angular momentum

24. The rotation group

25. Representations of the rotation group

26. Spherically symmetric operators

27. Representation of rotations by $2\times 2$ unitary matrices

28. Representation of the rotation group on a space of entire analytic functions of two complex variables

29. Uniqueness of the representations $D_j$

30. Representations of the rotation group on the space $L^2(S^2)$. Spherical functions

31. The radial Schrödinger equation

32. The hydrogen atom. The alkali metal atoms

33. Perturbation theory

34. The variational principle

35. Scattering theory. Physical formulation of the problem

36. Scattering of a onedimensional particle by a potential barrier

37. Physical meaning of the solutions $\psi _1$ and $\psi _2$

38. Scattering by a rectangular barrier

39. Scattering by a potential center

40. Motion of wave packets in a central force field

41. The integral equation of scattering theory

42. Derivation of a formula for the crosssection

43. Abstract scattering theory

44. Properties of commuting operators

45. Representation of the state space with respect to a complete set of observables

46. Spin

47. Spin of a system of two electrons

48. Systems of many particles. The identity principle

49. Symmetry of the coordinate wave functions of a system of two electrons. The helium atom

50. Multielectron atoms. Oneelectron approximation

51. The selfconsistent field equations

52. Mendeleev’s periodic system of the elements

53. Appendix. Lagrangian formulation of classical mechanics

The present volume has several desirable features. It speaks to mathematicians broadly, not merely practitioners of some narrow specialty. It faithfully explains physical ideas/concerns, rather than addresses the mathematician eager only to glean from physics a purely mathematical problem to attack. This book accomplishes its task as quickly as one could hope but still achieves interesting applications...Highly recommended.
D.V. Feldman, Choice