Angular Momentum Algebra

Canonical treatment: Angular Momentum Algebra owns the spectrum derivation, matrix realizations, physical examples, exercises, and references. This Toolkit page retains only the representation-theory prerequisites.

The Hermitian generators $J_i$ of rotations satisfy

$$[J_i,J_j]=i\hbar\epsilon_{ijk}J_k.$$

Mathematics texts often use dimensionless anti-Hermitian generators $T_i=-iJ_i/\hbar$, for which $[T_i,T_j]=\epsilon_{ijk}T_k$. Mixing these normalizations is a common source of missing factors of $i$ and $\hbar$.

Casimir and Weight Basis

The quadratic Casimir

$$J^2=J_x^2+J_y^2+J_z^2$$

commutes with every generator. In a finite-dimensional unitary irreducible representation of $\mathfrak{su}(2)$, one may choose

$$J^2\lvert j,m\rangle = \hbar^2j(j+1)\lvert j,m\rangle, \qquad J_z\lvert j,m\rangle = \hbar m\lvert j,m\rangle,$$

with $j=0,\tfrac12,1,\ldots$ and $m=-j,-j+1,\ldots,j$. Half-integer representations are representations of $SU(2)$ but do not descend to single-valued representations of $SO(3)$.

Ladder Identities

Define $J_\pm=J_x\pm iJ_y$. Then

$$[J_z,J_\pm]=\pm\hbar J_\pm, \qquad [J_+,J_-]=2\hbar J_z,$$

and

$$J_\mp J_\pm = J^2-J_z^2\mp\hbar J_z.$$

With the standard phase convention,

$$J_\pm\lvert j,m\rangle = \hbar\sqrt{j(j+1)-m(m\pm1)} \lvert j,m\pm1\rangle.$$

These identities are the minimal algebraic input for the physical ladder construction and for tensor-product coupling.

Routing

• Ladder Operators as Lie Algebra Tools gives the reusable root-and-weight pattern.
• SU(2) and SU(2) versus SO(3) explain the group-level distinction.
• Tensor-Product Representations leads to angular-momentum addition.
• Angular-Momentum Formula Sheet is the compact formula lookup.