# Symmetry and Quantum Mechanics

## Preview

## Book Description

Structured as a dialogue between a mathematician and a physicist, **Symmetry and Quantum Mechanics **unites the mathematical topics of this field into a compelling and physically-motivated narrative that focuses on the central role of symmetry.

Aimed at advanced undergraduate and beginning graduate students in mathematics with only a minimal background in physics, this title is also useful to physicists seeking a mathematical introduction to the subject. Part I focuses on spin, and covers such topics as Lie groups and algebras, while part II offers an account of position and momentum in the context of the representation theory of the Heisenberg group, along the way providing an informal discussion of fundamental concepts from analysis such as self-adjoint operators on Hilbert space and the Stone-von Neumann Theorem. Mathematical theory is applied to physical examples such as spin-precession in a magnetic field, the harmonic oscillator, the infinite spherical well, and the hydrogen atom.

## Table of Contents

**
**

**Physical Space.**Modeling space

Real linear operators and matrix groups

SO(3) is the group of rotations

**Spinor Space**Angular momentum in classical mechanics

Modeling spin

Complex linear operators and matrix groups

The geometry of

*SU*(2). The tangent space to the circle

*U*(1) =

*S*1

The tangent space to the sphere

*SU*(2) =

*S*3

The exponential of a matrix.

*SU*(2) is the universal cover of

*SO*(3)

Back to spinor space

**Observables and Uncertainty**Spin observables

The Lie algebra su(2)

Commutation relations and uncertainty

Some related Lie algebras

Warm-up: the Lie algebra u(1)

The Lie algebra sl2(C)

The Lie algebra u(2)

The Lie algebra gl2(C)

**Dynamics**Time-independent external fields

Time-dependent external fields

The energy-time uncertainty principle

Conserved quantities.

**Higher Spin.**Group representations.

Representations of

*SU*(2).

Lie algebra representations.

Representations of su(2)C = sl2(C).

Spin-

*s*particles.

Representations of

*SO*(3).

The so(3)-action

Comments about analysis.

**Multiple Particles. **Tensor products of representations.

The Clebsch-Gordan problem.

Identical particles—spin only.

**A One-dimensional World. **Position.

Momentum

The Heisenberg Lie algebra and Lie group

The meaning of the Heisenberg group action

Time-evolution

The free particle

The infinite square well

The simple harmonic oscillator

**A Three-dimensional World**Position

Linear momentum

The Heisenberg group

*H*3 and its algebra h3

Angular momentum

The Lie group

*G*=

*H*3 o

*SO*(3) and its Lie algebra g

Time-evolution

The free particle

The three-dimensional harmonic oscillator

Central potentials

The infinite spherical well

Two-particle systems

The Coulomb potential

Particles with spin

The hydrogen atom

Identical particles

**Towards a Relativistic Theory**Galilean relativity

Special relativity

*SL*2(C) is the universal cover of

*SO*+(1

*,*3)

The Dirac equation

**Appendices**Linear algebra

Vector spaces and linear transformations

Inner product spaces and adjoints

Multivariable calculus

Analysis

Hilbert spaces and adjoints

Some big theorems

Solutions to selected exercises

## Reviews

"In the preface to [this book] the author introduces the text as a ‘first course in quantum mechanics from the mathematical point of view’, whose main audience is ‘the advanced undergraduate student or beginning graduate student whose understanding of both physics and mathematics is just beginning to grow’. I would not hesitate to invite my colleagues who conduct undergraduate courses in quantum mechanics to the auditorium."

- Farhang Loran, Mathematical Reviews, August 2017