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Spin-spin interaction

  1. Nov 8, 2011 #1
    1. The problem statement, all variables and given/known data

    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?
     
  2. jcsd
  3. Nov 8, 2011 #2
    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.
     
  4. Nov 8, 2011 #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.)
     
  5. Nov 9, 2011 #4
    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
  6. Nov 10, 2011 #5

    vela

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    I'm not sure what you mean by this.

    It's just a multiplicative constant.
     
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