Angular Momentum

Phase 3: Advanced Quantum Systems

Syllabus Goal

General theory of angular momentum.

Commutation Relations

$$ [L_x, L_y] = i\hbar L_z $$ (and cyclic permutations).

Eigenvalues

Simultaneous eigenstates of $L^2$ and $L_z$: $$ L^2 | \ell, m \rangle = \hbar^2 \ell(\ell+1) | \ell, m \rangle $$ $$ L_z | \ell, m \rangle = \hbar m | \ell, m \rangle $$

Ladder Operators

$L_\pm = L_x \pm iL_y$. Used to step between $m$ states.