Another 2 questions in perturbation theory.

[MathematicalPhysicist]

Homework Statement

1. A particle of mass M is in a square well, subject to the potential:
$$V(x)= V0\theta(x-a/2)$$ for x in (0,a) and diverges elsewhere, where theta is heaviside step function.
In perturbation theory, find O(V0^2) correction to the energy and O(V0)to the eigenstate.

2. A particle is moving in a two dimensional harmonic oscillator with perturbation $$\delta V=\lambda x^2 y^2$$ For the ground state and the first excited state, find the first order correction to.

Homework Equations

The Attempt at a Solution

1. As I see it the hamiltonian has a perturbation in x in [a/2,a), which is V0.
, but from the equation for the second order correction to the energy which is:
$$\sum_{m doesn't equal n} \frac{|<m|V0|n>|^2}{E_n-E_m}$$, where $$E_k=\frac{(\hbar k)^2}{2M}+V_0$$, so that means the second order correction is zero cause <m|n> for m not equal n equlas zero, doesn't make sense to me, what am I missing?

For 2, for the first excited state I get that it's degenrate, cause $$E_{1,0}=E_{0,1}$$, not sure how to find for the first excited state its first order correction.

P.S
I sent an email to the TA with my questions, so far he hasn't replied, this is why I am asking here.

 

[diazona]

For #1, you can't just assume that $\left\vert \langle m \vert V_0 \vert n\rangle\right\vert = 0$ whenever $m \neq n$. Remember how you compute a matrix element:
$$\langle m \vert V_0 \vert n\rangle = \int\psi^{*}_m(x) V_0(x) \psi_n(x)\,\mathrm{d}x$$

 

[MathematicalPhysicist]

Ok thanks.
Can someone help me with question 2?

 

[MathematicalPhysicist]

bump, question 2?

 

[MathematicalPhysicist]

For #1, you can't just assume that $\left\vert \langle m \vert V_0 \vert n\rangle\right\vert = 0$ whenever $m \neq n$. Remember how you compute a matrix element:
$$\langle m \vert V_0 \vert n\rangle = \int\psi^{*}_m(x) V_0(x) \psi_n(x)\,\mathrm{d}x$$

I still get that it's zero.
What are eigenfunctions of this problem?

I think that they are:
$$\psi_n(x)=\sqrt(\frac{2}{a})sin(nx\pi/a)$$ but if it's so then <n|V|m>=0.