- #1

dreamspy

- 41

- 2

## 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:

[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]

## 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