Units#

Review & Summary#

Identical particles#

Indistinguishable quantum particles whose wave functions must be symmetric (bosons) or antisymmetric (fermions) under exchange.

Bosons / Fermions#

Integer-spin particles obeying \([a, a^\dagger] = 1\) / half-integer-spin particles obeying \(\{c, c^\dagger\} = 1\). Fermions satisfy the Pauli exclusion principle.

Second quantization#

Many-body formulation via creation \(a^\dagger\) and annihilation \(a\) operators acting on Fock space.

Angular momentum algebra#

\([J_i, J_j] = i\hbar \epsilon_{ijk} J_k\), \(J^2 |j,m\rangle = \hbar^2 j(j+1)|j,m\rangle\), \(J_z|j,m\rangle = \hbar m|j,m\rangle\).

Slater determinant#

Antisymmetrized \(N\)-fermion wave function: \(\Psi = \frac{1}{\sqrt{N!}}\det[\phi_i(\mathbf{r}_j)]\).

Anyons#

Particles in 2D with exchange phase \(e^{i\pi\theta}\) where \(\theta \notin \{0,1\}\). Arise in fractional quantum Hall systems.