Question

Determine the commutator Lz Ly.

Answer 1

Answer on Question #74740, Physics / Quantum Mechanics |

Determine the commutator Lz Ly.

Solution:

$$\begin{array}{l} \left[ L _ {z}, L _ {y} \right] = \left[ x p _ {y} - y p _ {x}, z p _ {x} - x p _ {z} \right] = \left[ x p _ {y}, z p _ {x} \right] - \left[ x p _ {y}, x p _ {z} \right] - \left[ y p _ {x}, z p _ {x} \right] + \left[ y p _ {x}, x p _ {z} \right] \\ = z p _ {x} [ x, p _ {x} ] - 0 - 0 + y p _ {z} [ p _ {x}, x ] \end{array}$$

Remembering that $[x, p_x] = -[p_x, x] = i\hbar$, we obtain

$$\left[ L _ {z}, L _ {y} \right] = i \hbar (z p _ {x} - y p _ {z}) = - i \hbar L _ {x}$$

Answer: $\left[L_z, L_y\right] = -i\hbar L_x$

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