# Perturbation Theory/Harmonic Oscillator

 P: 140 1. The problem statement, all variables and given/known data I am given the hamiltonian, where $$H^{^}_{0}$$ is that of the harmonic oscillator and the perturbation is (lambda)*(h-bar)*(omega)*[(lowering operator)^2 + (raising operator)^2]. I am asked to find the ground state, second-order approx. energy value. 2. Relevant equations Second order eigenenergy equation. 3. The attempt at a solution I have written out the whole hamiltonian. Do I need to expand the lowering-raising operators in terms of n? I am a bit lost on what to plug in to the second order equation for the two wave functions m and n (which sandwich the Hamiltonian operator).
 HW Helper PF Gold P: 3,442 You know that the "meat" of the sandwich is the perturbing Hamiltonian, $$H'=\hbar\omega(a_+^2+a_-^2)$$ Since you are looking for the second order correction to the ground state, one of the "breads", say the bra, should be <0|. Suppose you were to write the other "bread" (the ket) as |n>. What do you get when you operate on that with a+2 and a-2? What does the resulting ket need to be in order not to have a zero matrix element?
 P: 140 I have a question though: if the harmonic oscillator is (isotropic), how would two (lowering and raising) operators multiply each other, assuming they are from two different dimensions (e.g. x and y). Would you 'go up or down the ladder' the same way one usually would except plug in the different n-values for x and y, accordingly? In other words if you had $$a_{x} a_{y}$$, which are both lowering operators, would you go the typical route of bumping the eigenstate by |n-1>, |n-2>, etc. but plug in $$n_{x}$$ and $$n_{y}$$? The raising and lowering operators is still a fuzzy area with me.