Charged harmonic oscillator in an electric field

[bjogae]

Homework Statement

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

Homework Equations

The Attempt at a Solution

By completing the square i get
$$[-\frac{\hbar^2}{2m}\frac{d^2}{du^2}+\frac{1}{2}m \omega ^2u^2] \phi (u) = (E + \frac{q^2 \epsilon^2}{2m \omega ^2}) \phi (u)$$
where
$$u=x-\frac{q^2\epsilon^2}{2m\omega^2}$$.

Then usually for Hamiltonians of this kind the energy eigenvalues are
$$E_n=\hbar\omega(n+\frac{1}{2})$$
but how do I obtain them in this case? Or is this the right way to go?
Do i call
$$E + \frac{q^2 \epsilon^2}{2m \omega ^2}=E'$$
which would give me
$$E'_n=\hbar\omega(n+\frac{1}{2})$$
And how do I swich back to x?

 

[gabbagabbahey]

By completing the square i get
$$[-\frac{\hbar^2}{2m}\frac{d^2}{du^2}+\frac{1}{2}m \omega ^2u^2] \phi (u) = (E + \frac{q^2 \epsilon^2}{2m \omega ^2}) \phi (u)$$
where
$$u=x-\frac{q^2\epsilon^2}{2m\omega^2}$$.

That doesn't look quite right...