Solving Hamiltonian H=αL⋅S for L⋅S |(ls)jjz>

[atomicpedals]

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

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

 

[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.

 

[vela]

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

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

 

[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)?

 

[vela]

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?

I'm not sure what you mean by this.

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)?

It's just a multiplicative constant.