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## Homework Statement

I'm given that a harmonic oscillator is in a uniform gravitational field so that the potential energy is given by: [tex]V(x)=\frac{1}{2}m\omega^2x^2 - mgx[/tex], where the second term can be treated as a perturbation. I need to show that the first order correction to the energy of a stationary state is zero, and (the part I'm having trouble with) find the new energy eigenvalues to second order.

## Homework Equations

[tex]H^1_{nn} = <\psi_n|H^1\psi_n>[/tex]

[tex]E_n^2 = \Sigma_{j!=n}\frac{|H_{jn}|^2}{E_n^0 - E_j^0}[/tex]

## The Attempt at a Solution

For the first part, I just took -mgx as H^1 and used the first equation above, plugging in [tex]a_- - a_+[/tex] for x, so that I got [tex](a_- - a_+)\psi_n^0[/tex], which is zero (right?)

For the second part.. I'm not sure how to go about using the second equation. [tex]H_{jn} = <\psi_j^0|H\psi_n^0>[/tex].. but what do I use for H there? The same thing as above? And.. how do I go about doing a summation for these? I can't find any examples for the second order correction. I think En is just [tex]\frac{\hbar\omega}{2}[/tex], but Ej confuses me.

Thanks a ton for the help.

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