Spin-spin interaction

  • #1
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Homework Statement



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

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

2. The attempt at a solution

I'm highly tempted to start with
[tex]L \cdot S |(ls)jj_{z}> = \frac{\hbar^{2}}{2}[j(j+1)-l(l+1)-3/4]|(ls)jj_{z}>[/tex]
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

  • #2
208
7
Ok, so my first mistake was probably in my first crack at the solution; so I should probably have started with
[tex]L \cdot S |(ls)jj_{z}> = \frac{\hbar^{2}}{2}[j(j+1)-l(l+1)-s(s+1)]|(ls)jj_{z}>[/tex]
However, my initial self doubts still stand though.
 
  • #3
vela
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That's fine so far. Why are you doubting it's right?

(If you want to derive it, use the hint.)
 
  • #4
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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)?
 
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  • #5
vela
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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.
 

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