Perturbation theory

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Homework Statement



1. Considered the 2D harmonic oscillator potential,
V(x,y) = [tex]m\omega^{2}[/tex][tex]x^{2}/2[/tex]+[tex]m\omega^{2}[/tex][tex]y^{2}/2+ \lambda xy[/tex]
and showed that the energy eigenvalues could be found exactly. Now, treat this as a perturbation theory problem with perturbing Hamiltonian, [tex]H^{'}=\lambda xy[/tex]
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 [tex]\lambda[/tex] 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, [tex]H^{0} are [/tex]
[tex]\phi_{1} and \phi_{2}[/tex] with eigenvalues [tex]E_{1} and E_{2}[/tex] respectively. Now each of these states are subject to the perturbation: [tex]H^{'}[/tex]

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



Homework Equations



([tex]H^{0}[/tex]+[tex]\lambda H^{'}[/tex])([tex]\phi_{1}[/tex]+[tex]\lambda[/tex][tex]\phi_{2}[/tex]) = ([tex]E_{1}[/tex]+[tex]\lambda E_{2}[/tex])([tex]\phi_{1}[/tex]+[tex]\lambda[/tex][tex]\phi_{2}[/tex])

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.
 

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