Spin homework question - hard.

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
 
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?
 

MathematicalPhysicist

Gold Member
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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... :-)
 

samalkhaiat

Science Advisor
1,561
706
[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]
 
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] ?
 

samalkhaiat

Science Advisor
1,561
706
[QUOTE said:
dreamspy;2493358]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] ?
[/QUOTE]

That is a singlet state which has s = 0.
 

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