Perturbation Theory, exchange operator

[unscientific]

Homework Statement

Part (a): Find eigenvalues of X, show general relation of X and show X commutes with KE.
Part (b): Give conditions on V1, V2 and VI for X to commute with them.
Part (c): Write symmetric and antisymmetric wavefunctions. Find energies JD and JE.
Part (d): How are unperturbed and perturbed energy levels related?

Homework Equations

The Attempt at a Solution

Part (a)
The eigenvalues I found are $\pm 1$, which seems right.

No problem with this showing X commutes with K too.

Part(b)
I'm thinking X will commute with $x^2$, so this means that V₁ and V₂ must be functions of $x^2$.
As for V_I, I'm not sure.

Part (c)

For two particles A and B,
Symmetric wavefunction is:

$$\frac{1}{\sqrt 2} \left(|u,A>|v,B> + |v,A>|u,B>\right)$$

Anti-Symmetric wavefunction is:

$$\frac{1}{2} \left( |u,A>|v,B> - |v,A>|u,B>\right)$$

I'm not sure what $J_D$ and $J_E$ are. I'm guessing $J_E$ is the first order perturbed energy $<E|V_I|E>$?

Part (d)

The delta function means that $x_1 = x_2$, so the perturbed energy levels is the same as unperturbed? Not sure why either.

 

[unscientific]

I'm guessing $J_D$ is the interaction energy, $J_E$ is the first order perturbed energy. So this perturbation "lifts" the degeneracy.

Isn't $J_D = V_{|x_1-x_2|}$ while $J_E = <s+|V_I|s+>$ where $|s+>$ is the symmetric wavefunction?

How do I overlap this symmetric wavefunction: $\frac{1}{\sqrt 2} \left(|u,A>|v,B> + |v,A>|u,B>\right)$?
Since $u_{(x)}$ and $v_{(x)}$ are orthonormal states, can I assume $<u,A|v,A> = <u,B|v,B> = 0$?

Therefore $\langle\psi_s|\psi_s\rangle = \frac{1}{2}\int \langle u,A|u,A\rangle\langle v,B|v,B\rangle + \langle v,A|v,A\rangle\langle u,B|u,B\rangle d^3r = \int 2|u|^2|v|^2 d^3r$?

 

[unscientific]

bumpp