Spin-spin interaction

1. Nov 8, 2011

atomicpedals

1. The problem statement, all variables and given/known data

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?

2. Nov 8, 2011

atomicpedals

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.

3. Nov 8, 2011

vela

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

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

4. Nov 9, 2011

atomicpedals

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: Nov 9, 2011
5. Nov 10, 2011

vela

Staff Emeritus
I'm not sure what you mean by this.

It's just a multiplicative constant.