Symmetry and Quantum Mechanics

Scott Corry

Physical Space.
Modeling space
Real linear operators and matrix groups
SO(3) is the group of rotations

Spinor SpaceAngular 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) = S1
The tangent space to the sphere SU(2) = S3
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)

DynamicsTime-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 H3 and its algebra h3
Angular momentum
The Lie group G = H3 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 TheoryGalilean relativity
Special relativity
SL2(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