What Are the Eigenvalues and Eigenvectors of This Spin Hamiltonian?

[dreamspy]

Homework Statement

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:

$$\hat H = \gamma \hat S_3^{(1)} + \gamma \hat S_3^{(2)}$$

where $$\bf{S}^{(j)}$$ 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 $$u_s^{(1)}u_r^{(2)}$$ where$$r,s,=\pm\frac{1}{2}$$, and $$\hat S_3^{1}u_s^{1}=shu_s^{j}$$ and $$\hat S_3^{2}u_r^{2}=rhu_r^{j}$$

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

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

$$\hat H = \gamma \hat S_3^{(1)} + \gamma \hat S_3^{(2)} + \lambda \hat{\underline S}^{(1)}\cdot \lambda \hat{\underline S}^{(2)}$$

The Attempt at a Solution

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

(i)
Triplet:


$$u_{\frac{1}{2}}u_{\frac{1}{2}}$$
$$(u_{\frac{1}{2}}u_{-\frac{1}{2}}+u_{-\frac{1}{2}}u_{\frac{1}{2}})\frac{1}{\sqrt 2}$$
$$u_{-\frac{1}{2}}u_{-\frac{1}{2}}$$

$$\underlince{\hat S}^2 = s(s+1)$$

The Eigenvalues:

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

(ii)
Two spin operators:

$$\underline{\hat S}^{(1)}, \underline{\hat S}^{(2)}$$

$$\underline{\hat S}^{(1)}\cdot \underline{\hat S}^{(2)} = \frac{1}{2}(\underline{\hat S}^{2}-(\underline{\hat S}^{(1)})^2-(\underline{\hat S}^{(1)})^2)$$

$$\underline{\hat S} = \underline{\hat S}^{(1)} + \underline{\hat S}^{(2)}$$

$$\underline{\hat S}}$$ has eigenvalue $$s(s+1) \hbar ^2, s=0,1$$


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

 

[dreamspy]

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]

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]

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

remember

$$S_{1}^{2} = S_{2}^{2} = (1/2)[(1/2) + 1]$$

 

[dreamspy]

But what if we have $$u_{\frac{1}{2}}^1u_{-\frac{1}{2}}^2$$

How do we calculate $$\underline{\hat S}^2u_{\frac{1}{2}}^1u_{-\frac{1}{2}}^2$$ ?

 

[samalkhaiat]

dreamspy;2493358]But what if we have $$u_{\frac{1}{2}}^1u_{-\frac{1}{2}}^2$$

How do we calculate $$\underline{\hat S}^2u_{\frac{1}{2}}^1u_{-\frac{1}{2}}^2$$ ?

[/QUOTE]

That is a singlet state which has s = 0.