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Homework Help: Spin homework question - hard.

  1. Dec 15, 2009 #1
    1. The problem statement, all variables and given/known data

    I'm struggling with this question here from my QM class. I have read all my material on Spin (both Griffiths, and a chapter in an Icelandic book). I'we done some problems, but I really have no Idea where to start with this one. It goes something like this:

    Two particles have spin 1/2 and are stationary, but their spins interact with this Hamilton operator:

    [tex]\hat H = \gamma \hat S_3^{(1)} + \gamma \hat S_3^{(2)} [/tex]

    where [tex]\bf{S}^{(j)} [/tex] is the spin operator for particle j, and j=1,2.

    As a basis in the state space (hope that's the right word) you can f.x. take [tex]u_s^{(1)}u_r^{(2)}[/tex] where[tex] r,s,=\pm\frac{1}{2}[/tex], and [tex]\hat S_3^{1}u_s^{1}=shu_s^{j}[/tex] and [tex]\hat S_3^{2}u_r^{2}=rhu_r^{j}[/tex]

    Questions:
    (i) Find the eigenvalues and eigenvektors of the Hamilton operator.

    (ii) How would the result be if we used this Hamilton operator instead:

    [tex]\hat H = \gamma \hat S_3^{(1)} + \gamma \hat S_3^{(2)} + \lambda \hat{\underline S}^{(1)}\cdot \lambda \hat{\underline S}^{(2)} [/tex]

    3. The attempt at a solution

    Now here is a quick solution I got from my teacher:

    (i)
    Triplet:


    [tex]u_{\frac{1}{2}}u_{\frac{1}{2}}[/tex]
    [tex](u_{\frac{1}{2}}u_{-\frac{1}{2}}+u_{-\frac{1}{2}}u_{\frac{1}{2}})\frac{1}{\sqrt 2}[/tex]
    [tex]u_{-\frac{1}{2}}u_{-\frac{1}{2}}[/tex]

    [tex]\underlince{\hat S}^2 = s(s+1)[/tex]

    The Eigenvalues:

    [tex](\gamma S_3^{(1)}-\gamma S_3^{(2)})u_{\frac{1}{2}}u_{\frac{1}{2}} = \gamma \hbar(s+r)u_{\frac{1}{2}}u_{\frac{1}{2}}[/tex]
    (the others should follow the same procedure)

    (ii)
    Two spin operators:

    [tex]\underline{\hat S}^{(1)}, \underline{\hat S}^{(2)}[/tex]

    [tex]\underline{\hat S}^{(1)}\cdot \underline{\hat S}^{(2)} = \frac{1}{2}(\underline{\hat S}^{2}-(\underline{\hat S}^{(1)})^2-(\underline{\hat S}^{(1)})^2)[/tex]

    [tex]\underline{\hat S} = \underline{\hat S}^{(1)} + \underline{\hat S}^{(2)}
    [/tex]

    [tex]\underline{\hat S}}[/tex] has eigenvalue [tex]s(s+1) \hbar ^2, s=0,1[/tex]


    Now I almost have no clue on what's going on here.

    Now I suppose the part in the Triplet section, is all possible linear combinations of the u vektors. And the eigenvalue can be read from the right side of the formula below. But could anyone care to comment on this? I'm standing on very shaky ground here :) This is the solution my teacher gave us, nobody has a clue what's going on, and were taking the exam tomorrow :)


    Thanks in advance for any comments!
    Frímann
     
  2. jcsd
  3. Dec 15, 2009 #2
    My basic quesion is probably, how do I choose the part in the Triplet section, are those the eigenfunctions? If not then how would I find them?
     
  4. Dec 15, 2009 #3

    MathematicalPhysicist

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    Gold Member

    Well if I am not mistaken for two fermions the eigenstate should be anti symmetrical, i.e singlet and not triplet which is symmetric.

    But I myself in a shaky ground... :-)
     
  5. Dec 15, 2009 #4

    samalkhaiat

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    Science Advisor

    [tex]\underline{\hat S}^{(1)}\cdot \underline{\hat S}^{(2)} = \frac{1}{2}(\underline{\hat S}^{2}-(\underline{\hat S}^{(1)})^2-(\underline{\hat S}^{(1)})^2) = (1/2)[s(s+1) - (3/2)]
    [/tex]

    remember

    [tex]S_{1}^{2} = S_{2}^{2} = (1/2)[(1/2) + 1][/tex]
     
  6. Dec 15, 2009 #5
    But what if we have [tex]u_{\frac{1}{2}}^1u_{-\frac{1}{2}}^2[/tex]

    How do we calculate [tex]\underline{\hat S}^2u_{\frac{1}{2}}^1u_{-\frac{1}{2}}^2[/tex] ?
     
  7. Dec 15, 2009 #6

    samalkhaiat

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    Science Advisor

    [/QUOTE]

    That is a singlet state which has s = 0.
     
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