**Student Mathematical Library**

Volume: 47;
2009;
234 pp;
Softcover

MSC: Primary 81;

Print ISBN: 978-0-8218-4699-5

Product Code: STML/47

List Price: $41.00

Individual Price: $32.80

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**Electronic ISBN: 978-1-4704-1633-1
Product Code: STML/47.E**

List Price: $41.00

Individual Price: $32.80

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#### Supplemental Materials

# Lectures on Quantum Mechanics for Mathematics Students

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*L. D. Faddeev; O. A. Yakubovskiĭ*

with an appendix by Leon Takhtajan

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

# Table of Contents

## Lectures on Quantum Mechanics for Mathematics Students

- Cover Cover11 free
- Title page iii5 free
- Contents v8 free
- Preface ix12 free
- Preface to the English edition xi14 free
- The algebra of observables in classical mechanics 116 free
- States 621
- Liouville’s theorem, and two pictures of motion in classical mechanics 1328
- Physical bases of quantum mechanics 1530
- A finite-dimensional model of quantum mechanics 2742
- States in quantum mechanics 3247
- Heisenberg uncertainty relations 3651
- Physical meaning of the eigenvalues and eigenvectors of observables 3954
- Two pictures of motion in quantum mechanics. The Schrödinger equation. Stationary states 4459
- Quantum mechanics of real systems. The Heisenberg commutation relations 4964
- Coordinate and momentum representations 5469
- “Eigenfunctions” of the operators 𝑄 and 𝑃 6075
- The energy, the angular momentum, and other examples of observables 6378
- The interconnection between quantum and classical mechanics. Passage to the limit from quantum mechanics to classical mechanics 6984
- One-dimensional problems of quantum mechanics. A free one-dimensional particle 7792
- The harmonic oscillator 8398
- The problem of the oscillator in the coordinate representation 87102
- Representation of the states of a one-dimensional particle in the sequence space 𝑙₂ 90105
- Representation of the states for a one-dimensional particle in the space 𝒟 of entire analytic functions 94109
- The general case of one-dimensional motion 95110
- Three-dimensional problems in quantum mechanics. A three-dimensional free particle 103118
- A three-dimensional particle in a potential field 104119
- Angular momentum 106121
- The rotation group 108123
- Representations of the rotation group 111126
- Spherically symmetric operators 114129
- Representation of rotations by 2×2 unitary matrices 117132
- Representation of the rotation group on a space of entire analytic functions of two complex variables 120135
- Uniqueness of the representations 𝐷ⱼ 123138
- Representations of the rotation group on the space 𝐿²(𝑆²). Spherical functions 127142
- The radial Schrödinger equation 130145
- The hydrogen atom. The alkali metal atoms 136151
- Perturbation theory 147162
- The variational principle 154169
- Scattering theory. Physical formulation of the problem 157172
- Scattering of a one-dimensional particle by a potential barrier 159174
- Physical meaning of the solutions 𝜓₁ and 𝜓₂ 164179
- Scattering by a rectangular barrier 167182
- Scattering by a potential center 169184
- Motion of wave packets in a central force field 175190
- The integral equation of scattering theory 181196
- Derivation of a formula for the cross-section 183198
- Abstract scattering theory 188203
- Properties of commuting operators 197212
- Representation of the state space with respect to a complete set of observables 201216
- Spin 203218
- Spin of a system of two electrons 208223
- Systems of many particles. The identity principle 212227
- Symmetry of the coordinate wave functions of a system of two electrons. The helium atom 215230
- Multi-electron atoms. One-electron approximation 217232
- The self-consistent field equations 223238
- Mendeleev’s periodic system of the elements 226241
- Lagrangian formulation of classical mechanics 231246
- Back Cover Back Cover1252

#### Readership

Undergraduate and graduate students interested in learning the basics of quantum 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