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

We're supposed to consider the Hamiltonian for the simple harmonic oscillator:

[tex]\hat{H}_{0} = \hat{p}^{2}/2m + m\omega^2\hat{x}^2/2[/tex]

With a perturbation, so that [tex]\hat{H} = \hat{H}_0 + \hat{H}' [/tex], where [tex]\hat{H}' = F\hat{x}[/tex]

I've already solved for the first and second order energy corrections, as well as the first order correction to the wave function. The last part of the question is to solve for the exact energies using the change of variables [tex]x' \equiv x + F/m\omega^2[/tex]

2. Relevant equations

See above

3. The attempt at a solution

When I substitute for x in the perturbed Hamiltonian, I get

[tex]\hat{H} = \hat{p}^2/2m + m\omega^2\hat{x'}^2/2 - F^2/2m\omega^2[/tex]

Which is of the same form as the unperturbed Hamiltonian, except for a constant, which would shift the energies down by the constant compared to the unperturbed case. However, when I solved for the second order correction to the energies, I got that there wasn't a constant shift, but instead the spacing of the energy levels increased by [tex]F^2(n+1/2)/m\omega^2[/tex] Since the energy correction looks a lot like the extra constant that was introduced to the Hamiltonian, I'm inclined to think that my reasoning is incorrect somewhere, and that I should get that the exact energies are equal to the 0th plus 2nd order energies, but that's just a guess.

Does anyone know where I'm going wrong?

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# Homework Help: Constant force perturbation of the quantum SHO

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