# Quantum harmonical oscillator with electric field

#### Chen

Hi,

I have a particle of mass m and charge q, which is located in the potential of an harmonic oscillator and also subject to a constant electric field. The Hamiltonian is given as:

$$H = \frac{p^2}{2m} + \frac{1}{2}m \omega ^2 x^2 - q E' x$$

And I need to find a change of variables from x to u, so that the eigenvalue equation:

$$H \phi (x) = E \phi (x)$$

Becomes:

$$[-\frac{h^2}{2m}\frac{d^2}{du^2}+\frac{1}{2}m \omega ^2u^2] \phi (u) = (E + \frac{q^2 E'^2}{2m \omega ^2}) \phi (u)$$

(It's an h-bar there, of course.) I don't even know where to start. I tried plugging u(x) into the original eigenvalue equation and find some constraint on u from there, to no avail.

Thanks

Last edited:
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#### George Jones

Staff Emeritus
Gold Member
Complete the square on the last 2 terms in the Hamiltonian, and the transformation might become a bit more obvious.

Regards,
George

Doh... thanks!

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