# Perturbation theory

1. Oct 21, 2009

### ZoroP

1. The problem statement, all variables and given/known data

1. Considered the 2D harmonic oscillator potential,
V(x,y) = $$m\omega^{2}$$$$x^{2}/2$$+$$m\omega^{2}$$$$y^{2}/2+ \lambda xy$$
and showed that the energy eigenvalues could be found exactly. Now, treat this as a perturbation theory problem with perturbing Hamiltonian, $$H^{'}=\lambda xy$$
a) Find the first and second-order shifts in the energy of the ground state.
b) Series expand the exact energy level of the ground state in powers of $$\lambda$$ and show that the first and second order terms agree with your calculations from part a)

2. In a 2-D state space, the eigenstates of the Hamiltonian, $$H^{0} are$$
$$\phi_{1} and \phi_{2}$$ with eigenvalues $$E_{1} and E_{2}$$ respectively. Now each of these states are subject to the perturbation: $$H^{'}$$

If $$E_{1} \neq E_{2}$$, what are the new energy levels according to first-order perturbation theory?

2. Relevant equations

($$H^{0}$$+$$\lambda H^{'}$$)($$\phi_{1}$$+$$\lambda$$$$\phi_{2}$$) = ($$E_{1}$$+$$\lambda E_{2}$$)($$\phi_{1}$$+$$\lambda$$$$\phi_{2}$$)

3. The attempt at a solution

For 1. part b, i dont know what this question asks me to do. We can use the formulas for part a, so does part b ask us to derive the formula from the beginning with H' ? I get the solution for part a, such as E1=0 and E2=-P*(2n+1) where P is some constant. I dont know whether they are right or not.

For 2. do we start with the equation above?

Thanks a lot.