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