Perturbation Theory: 2D Harmonic Oscillator & Energy Levels

[ZoroP]

Homework Statement

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?

Homework Equations

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

The Attempt at a Solution

For 1. part b, i don't 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 don't know whether they are right or not.

For 2. do we start with the equation above?

Thanks a lot.

 

[Jhero]

Dear fellow scientist,

Thank you for your forum post. It seems like you are working on a problem related to perturbation theory and eigenstates of the 2D harmonic oscillator potential. I would be happy to help you with your questions.

For part b of question 1, it seems like you are being asked to show that the first and second-order energy shifts you calculated in part a agree with the series expansion of the exact energy level of the ground state in powers of \lambda. To do this, you can start with the formula you provided in your "Homework Equations" section and use it to expand the energy level in powers of \lambda. Then, compare the coefficients of the first and second-order terms with the ones you calculated in part a to show that they are the same.

As for your solution for part a, I am not able to confirm whether it is correct without seeing your calculations. However, it seems like you have the right approach and I encourage you to double-check your calculations to make sure they are accurate.

For part 2, yes, you can start with the equation you provided and use it to calculate the new energy levels according to first-order perturbation theory. Remember that in this case, the eigenvalues E_{1} and E_{2} are different, so you will have to use different equations for each state.

I hope this helps. Let me know if you have any further questions or if you would like me to clarify anything. Good luck with your calculations!