Question 705027

SCQMEDIUM

If $\vec{L}$ is the orbital angular momentum operator and $\vec{\sigma}$ are the Pauli matrices, which of the following operators commutes with $\vec{\sigma} \cdot \vec{L}$ ?

(A)

$\vec{L} - \frac{\hbar}{2}\vec{\sigma}$

(B)

$\vec{L} + \frac{\hbar}{2}\vec{\sigma}$

(C)

$\vec{L} + \hbar\vec{\sigma}$

(D)

$\vec{L} - \hbar\vec{\sigma}$

Detailed Solution

The total angular momentum $\vec{J} = \vec{L} + \vec{S} = \vec{L} + \frac{\hbar}{2}\vec{\sigma}$ . The spin-orbit coupling term is proportional to $\vec{L} \cdot \vec{S}$ . Since the total angular momentum commutes with the spin-orbit interaction, $\vec{L} + \frac{\hbar}{2}\vec{\sigma}$ commutes with $\vec{\sigma} \cdot \vec{L}$ .

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