## Related questions with answers

The spin-orbit energy is proportional to $\mathbf{S} \cdot \mathbf{L}$. To evaluate $\mathbf{S} \cdot \mathbf{L}$, note that since $\mathbf{J}=\mathbf{S}+\mathbf{L}$,

$J^2=(\mathbf{S}+\mathbf{L}) \cdot(\mathbf{S}+\mathbf{L})=S^2+L^2+2 \mathbf{S} \cdot \mathbf{L}$

You can solve this for $\mathbf{S} \cdot \mathbf{L}$ and then put in the values for $J^2, S^2$, and $L^2$. [For instance, $J^2=j(j+1) \hbar^2$ where $j=l \pm \frac{1}{2}$.] Show that if $j=l+\frac{1}{2}$ then $\mathbf{S} \cdot \mathbf{L}=$ $l^2 / 2$, and if $j=l-\frac{1}{2}$, then $\mathbf{S} \cdot \mathbf{L}=-(l+1) \hbar^2 / 2$. Use these answers to prove that the spin-orbit splitting increases with increasing $l$, as indicated in the above figure.

Solution

VerifiedThe spin-orbit energy comes from the relative orientation of the orbital angular momentum $\bold{L}$ to the spin $\bold{S}$. This energy is proportional to $\bold{S}\cdot\bold{L}$ and causes a splitting of the orbital $l$. The goal of this exercise is to find $\bold{S}\cdot\bold{L}$.

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