# Spin-spin interaction

## Homework Statement

Consider the Hamiltonian
$$H=\alpha L \cdot S$$
Where L denotes an angular momentum with quantum number l and S a spin with quantum number s.

Work out
$$L \cdot S |(ls)jj_{z}>$$
direction. Hint: expand (L+S)2 and go from there.

2. The attempt at a solution

$$L \cdot S |(ls)jj_{z}> = \frac{\hbar^{2}}{2}[j(j+1)-l(l+1)-3/4]|(ls)jj_{z}>$$
except I'm not sure that really buys me much. If I'm on the right track, how do I then handle the ket?

## Answers and Replies

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Ok, so my first mistake was probably in my first crack at the solution; so I should probably have started with
$$L \cdot S |(ls)jj_{z}> = \frac{\hbar^{2}}{2}[j(j+1)-l(l+1)-s(s+1)]|(ls)jj_{z}>$$
However, my initial self doubts still stand though.

vela
Staff Emeritus
Homework Helper
That's fine so far. Why are you doubting it's right?

(If you want to derive it, use the hint.)

I'm doubting it almost out of habit, lately I've started down the wrong path more often than not.

So, given that I've actually got this one going correctly, what do I do with the information in the ket?

And as a more in-depth question; what does the alpha term do to things when I want to calculate the energy (full disclosure: the follow up question actually is to calculate the energy spectrum, where I would tend to want to start from the TISE)?

Last edited:
vela
Staff Emeritus