Try the fastest way to create flashcards
Question

# 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

Verified
Step 1
1 of 9

The 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 $\bold{S}\cdot\bold{L}$.

## Recommended textbook solutions #### Physics for Scientists and Engineers: A Strategic Approach with Modern Physics

4th EditionISBN: 9780133942651 (8 more)Randall D. Knight
3,508 solutions #### Modern Physics for Scientists and Engineers

2nd EditionISBN: 9780138057152 (3 more)Chris D. Zafiratos, John R. Taylor, Michael A. Dubson
860 solutions #### Mathematical Methods in the Physical Sciences

3rd EditionISBN: 9780471198260 (1 more)Mary L. Boas
3,355 solutions #### Fundamentals of Physics

10th EditionISBN: 9781118230718 (3 more)David Halliday, Jearl Walker, Robert Resnick
8,971 solutions