# Angular Momentum Commutation Rules

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## Theorem

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

Then:

\(\displaystyle \sqbrk {J_x, J_y}\) | \(=\) | \(\displaystyle i J_z\) | |||||||||||

\(\displaystyle \sqbrk {J_y, J_z}\) | \(=\) | \(\displaystyle i J_x\) | |||||||||||

\(\displaystyle \sqbrk {J_z, J_x}\) | \(=\) | \(\displaystyle i J_y\) |

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

## Proof

## Sources

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