Question: Evaluate the following commutators :

a)$\left[L·S,L\right]$

b)$\mathbf{[}\mathbf{L}\mathbf{·}\mathbf{S}\mathbf{,}\mathbf{S}\mathbf{]}$

c)role="math" localid="1658226147021" $\mathbf{}\mathbf{[}\mathbf{L}\mathbf{·}\mathbf{S}\mathbf{,}\mathbf{J}\mathbf{]}$

d)$\mathbf{[}\mathbf{L}\mathbf{·}\mathbf{S}\mathbf{,}{\mathbf{L}}^{\mathbf{2}}\mathbf{]}$

e)$\mathbf{[}\mathbf{L}\mathbf{·}\mathbf{S}\mathbf{,}{\mathbf{S}}^{\mathbf{2}}\mathbf{]}$

f)$\mathbf{[}\mathbf{L}\mathbf{·}\mathbf{S}\mathbf{,}{\mathbf{J}}^{\mathbf{2}}\mathbf{]}$

Hint: L and S satisfy the fundamental commutation relations for angular momentum (Equations 4.99 and 4.134 ), but they commute with each other.

$$\begin{gathered} \left[L_X,L_Y\right]\mathbf{=}{\mathbf{ihL}}_{\mathbf{z}}\mathbf{;}\mathbf{[}{\mathbf{L}}_{\mathbf{y}}\mathbf{,}{\mathbf{L}}_{\mathbf{z}}\mathbf{]}\mathbf{=}{\mathbf{ihL}}_{\mathbf{x}}\mathbf{;}\mathbf{[}{\mathbf{L}}_{\mathbf{z}}\mathbf{,}{\mathbf{L}}_{\mathbf{x}}\mathbf{]}\mathbf{=}{\mathbf{ihL}}_{\mathbf{y}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{4}\mathbf{.}\mathbf{99} \\ \mathbf{[}{\mathbf{S}}_{\mathbf{X}}\mathbf{,}{\mathbf{S}}_{\mathbf{Y}}\mathbf{]}\mathbf{=}{\mathbf{ihS}}_{\mathbf{z}}\mathbf{;}\mathbf{[}{\mathbf{S}}_{\mathbf{y}}\mathbf{,}{\mathbf{S}}_{\mathbf{z}}\mathbf{]}\mathbf{=}{\mathbf{ihS}}_{\mathbf{x}}\mathbf{;}\mathbf{[}{\mathbf{S}}_{\mathbf{z}}\mathbf{,}{\mathbf{S}}_{\mathbf{x}}\mathbf{]}\mathbf{=}{\mathbf{ihS}}_{\mathbf{y}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{4}\mathbf{.}\mathbf{134} \end{gathered}$$

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