[PhD Qualifier] Spin eigenfunctions

[confuted]

Homework Statement

Consider two identical particles of mass m and spin 1/2. They interact via a potential given by
$$V=\frac{g}{r}\hat{\sigma}_1\cdot\hat{\sigma}_2$$

where g>0 and $$\hat{sigma}_j$$ are Pauli spin matrices which operate on the spin of particle j.

a) Construct the spin eigenfunctions for the two particle states. What is the expectation value of V for each of these states?

b) Give eigenvalues of all the bound states

Homework Equations

The Attempt at a Solution

I'm really not sure where to start. I'm used to the Hamiltonian taking the form
$$H=\frac{\hat{p}^2}{2m}+V$$
but I don't see how I can apply that at all. Is this easier than I'm making it?

I thought I was relatively good at Q.M... hints?

 

[Mute]

Your wavefunction is going to be a product of a spatial part and a spin part, the spin part being a vector. The sigma matrices in the potential energy term will act only on the spin part, which is going to be one of the spin triplets or the spin singlet. To deal with them, it's best to rewrite the dot product:

$$(\mathbf{\sigma}_1 + \mathbf{\sigma}_2)^2 = \mathbf{\sigma}_1^2 + \mathbf{\sigma}_2^2 + 2\mathbf{\sigma}_1 \cdot \mathbf{\sigma}_2$$

Solve for the dot product, then recall that $\sigma^2 = 1$, and the sum squared is the total spin momentum operator squared (well, without the factor of $(\hbar/2)^2$). When calculating the expected value of V, there will be two possible cases depending on the total spin momentum.

Does that help you get started?

 

[SonOfOle]

I've got the same question, and I'm still stuck after reading your reply Mute. Could you by chance elaborate more (or, if you've got it confuted, could you post what you know)?

Thanks,

 

[Mute]

Basically, your wavefunction should, I believe, look like $\psi(\mathbf{r})\left|s m_s \right>$, where $\psi(\mathbf{r})$ is the spatial part and $\left|s m_s \right>$ is the spin vector part (a triplet or singlet state). Then,

$$\mathcal{H}\psi(\mathbf{r})\left|s m_s \right> = E\psi(\mathbf{r})\left|s m_s \right> \Rightarrow \left(\frac{\hat{p}^2}{2m}+\frac{g}{r}\hat{\sigma}_1\cdot\hat{\sigma}_2 - E \right)\psi(\mathbf{r})\left|s m_s \right> = 0$$

The potential energy operator will operate on the spin vector, replacing the sigma dot product with a number $\lambda$ (which depends on the total spin number s). You then solve the equation

$$\left(\frac{\hat{p}^2}{2m}+\frac{g\lambda}{r} - E \right)\psi(\mathbf{r}) = 0$$

for the spatial part. You can then calculate the expected value of the potential.

I think this is the way to do it. I'm not 100% certain, but it makes sense to me. Then again, I haven't actually solved this problem myself.

If that's wrong, then I guess the way to do it would be to use the matrix form and write the wave function as a column vector with components $\psi_A(\mathbf{r})$ and $\psi_B(\mathbf{r})$, then rewrite the spin operators as above and write the total spin operator in matrix form, and solve the corresponding coupled equations for $\psi_A(\mathbf{r})$ and $\psi_B(\mathbf{r})$.

 

[badphysicist]

Looking at the operator, $$\hat{\sigma}_1\cdot\hat{\sigma}_2$$, you can decide to try your basic up and down single electron spin statse put together in various forms and see what happens when you act on them with the aforementioned operator. You want to make sure that you get back the same state you put in when you operate upon the state with that operator - no negative sign or anything. You will get the triplet states and the singlet state that satisfy this. It seems like this is the way to go, but maybe I'm wrong. The rest of the potential is just that of a hydrogen atom essentially. Just make sure that your spatial asymmetric states go with the symmetric spin states and vice versa since these are fermions.

 

[will.c]

Since they are identical fermions (as opposed to, for example, an electron and a proton for which Mute's explanation seems satisfactory to me) doesn't the exclusion principle have to be invoked in some way? I mean, in two of the triplet states the spins match. Is this problem avoided by something else in the way the problem is worded?

 

[badphysicist]

Yes, the Pauli exclusion principle is involved. That is why either the spatial part or the spin part of the wave function has to be asymmetric (but not both). If both of the spins are up for instance (one of the triplet states), the spatial function has to be asymetrical (they don't have the exact same n,l and m).