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Lectures on Quantum Mechanics for Mathematics Students

L. D. Faddeev Steklov Mathematical Institute, St. Petersburg, Russia
O. A. Yakubovskiĭ St. Petersburg University, St. Petersburg, Russia
with an appendix by Leon Takhtajan
Available Formats:
Softcover ISBN: 978-0-8218-4699-5
Product Code: STML/47
List Price: $44.00 Individual Price:$35.20
Electronic ISBN: 978-1-4704-1633-1
Product Code: STML/47.E
List Price: $41.00 Individual Price:$32.80
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List Price: $66.00 Click above image for expanded view Lectures on Quantum Mechanics for Mathematics Students L. D. Faddeev Steklov Mathematical Institute, St. Petersburg, Russia O. A. Yakubovskiĭ St. Petersburg University, St. Petersburg, Russia with an appendix by Leon Takhtajan Available Formats:  Softcover ISBN: 978-0-8218-4699-5 Product Code: STML/47  List Price:$44.00 Individual Price: $35.20  Electronic ISBN: 978-1-4704-1633-1 Product Code: STML/47.E  List Price:$41.00 Individual Price: $32.80 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$66.00
• Book Details

Student Mathematical Library
Volume: 472009; 234 pp
MSC: 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.

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 finite-dimensional 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. One-dimensional problems of quantum mechanics. A free one-dimensional particle
• 16. The harmonic oscillator
• 17. The problem of the oscillator in the coordinate representation
• 18. Representation of the states of a one-dimensional particle in the sequence space $l_2$
• 19. Representation of the states for a one-dimensional particle in the space $\mathcal {D}$ of entire analytic functions
• 20. The general case of one-dimensional motion
• 21. Three-dimensional problems in quantum mechanics. A three-dimensional free particle
• 22. A three-dimensional 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 one-dimensional 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 cross-section
• 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. Multi-electron atoms. One-electron approximation
• 51. The self-consistent field equations
• 52. Mendeleev’s periodic system of the elements
• 53. Appendix. Lagrangian formulation of classical mechanics

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 472009; 234 pp
MSC: 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.

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 finite-dimensional 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. One-dimensional problems of quantum mechanics. A free one-dimensional particle
• 16. The harmonic oscillator
• 17. The problem of the oscillator in the coordinate representation
• 18. Representation of the states of a one-dimensional particle in the sequence space $l_2$
• 19. Representation of the states for a one-dimensional particle in the space $\mathcal {D}$ of entire analytic functions
• 20. The general case of one-dimensional motion
• 21. Three-dimensional problems in quantum mechanics. A three-dimensional free particle
• 22. A three-dimensional 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 one-dimensional 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 cross-section
• 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. Multi-electron atoms. One-electron approximation
• 51. The self-consistent 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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