Angular Momentum Commutation Rules

Theorem

Let $J_x$, $J_y$ and $J_z$ denote the angular momentum operators.

Then:

\(\ds \sqbrk {J_x, J_y}\)

\(=\)

\(\ds i J_z\)

\(\ds \sqbrk {J_y, J_z}\)

\(=\)

\(\ds i J_x\)

\(\ds \sqbrk {J_z, J_x}\)

\(=\)

\(\ds i J_y\)

where $\sqbrk {\, \cdot, \cdot \,}$ denotes the commutator operator.

Proof

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Sources

• 1966: Harry J. Lipkin: Lie Groups for Pedestrians (2nd ed.): $\S 1$: Introduction