2 spin 1/2 particles in magnetic field

[oww]

Homework Statement

See pdf file: http://www.2shared.com/file/3469465/52e02653/q3_online.html

Its the third problem I'm having trouble with. I'm pretty sure I have the two first questions correct.

Homework Equations

1)$$H^{'}_{\uparrow\downarrow,\downarrow\uparrow}=<\left|\uparrow\downarrow\right\rangle\left|H^'\right|\left|\downarrow\uparrow\right\rangle$$

The Attempt at a Solution

My idea was to compute the quantity given by 1), but I am not sure how to manage $$H^{'} =-\beta S^{(1)}.S^{(2)}=-\beta(S_x ^{(1)}S_x ^{(2)},S_y ^{(1)}S_y ^{(2)},S_z ^{(1)}S_z ^{(2)})$$

Should I try and find $$S_y ^{(1)}$$, $$S_y ^{(2)}$$ and the same for the z-spinor matrices. Or is there a smarter way of doing this?

Looking forward to all the help i can get. Oistein

 

[Mute]

You've written

$$H^{single-quote} =-\beta S^{(1)}.S^{(2)}=-\beta(S_x ^{(1)}S_x ^{(2)},S_y ^{(1)}S_y ^{(2)},S_z ^{(1)}S_z ^{(2)})$$

which you appear to have written as a vector, which it isn't:

$$\mathcal{H} = - \beta \mathbf{S}_1 \cdot \mathbf{S}_2 = -\beta \left[S_{x1}S_{x2} + S_{y1}S_{y2} + S_{z1}S_{z2}\right]$$

Now, you might not want to solve the hamiltonian in that form. Instead, look at:

$$(\mathbf{S}_1 + \mathbf{S}_2)^2 = S_1^2 + S_2^2 + 2\mathbf{S}_1 \cdot \mathbf{S}_2$$

Solve for the dot product and substitute that into the Hamiltonian. Having done that, your hamiltonian is expressed in terms of the total spin operator and the spin operators for particle 1 and particle two. These last two are proportional to the Pauli spin matrices, and you should know that $\mathbf{\sigma}^2 = 1$, so S_1^2 and S_2^2 just evaluate to the square of the proportionality factors. So, the only operator whose value you don't know before acting on it with your states to get the matrix element is the total spin operator.

 

[Mute]

Blah, time limit to edit has expired before I was finished editing. What's left to say is that your matrix is 4x4, so to show that the hamiltonian is non-diagonal in your basis states, you just need to find a non-zero non-diagonal entry. The problem seems to tell you which element to check. Do that and you should be done.