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**1. Homework Statement**

A charged harmonic oscillator is placed in an external electric field [tex]\epsilon[/tex] i.e. its hamiltonian is [tex] H = \frac{p^2}{2m} + \frac{1}{2}m \omega ^2 x^2 - q \epsilon x [/tex] Find the eigenvalues and eigenstates of energy

**2. Homework Equations**

**3. The Attempt at a Solution**

By completing the square i get

[tex] [-\frac{\hbar^2}{2m}\frac{d^2}{du^2}+\frac{1}{2}m \omega ^2u^2] \phi (u) = (E + \frac{q^2 \epsilon^2}{2m \omega ^2}) \phi (u) [/tex]

where

[tex]u=x-\frac{q^2\epsilon^2}{2m\omega^2}[/tex].

Then usually for Hamiltonians of this kind the energy eigenvalues are

[tex]E_n=\hbar\omega(n+\frac{1}{2})[/tex]

but how do I obtain them in this case? Or is this the right way to go?

Do i call

[tex]E + \frac{q^2 \epsilon^2}{2m \omega ^2}=E'[/tex]

which would give me

[tex]E'_n=\hbar\omega(n+\frac{1}{2})[/tex]

And how do I swich back to x?