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## 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 V

_{1}and V

_{2}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:

[tex]\frac{1}{\sqrt 2} \left(|u,A>|v,B> + |v,A>|u,B>\right)[/tex]

Anti-Symmetric wavefunction is:

[tex]\frac{1}{2} \left( |u,A>|v,B> - |v,A>|u,B>\right)[/tex]

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.