External field applied to Harmonic Oscillator

[unscientific]

Homework Statement

For a particle of charge $q$ in a potential $\frac{1}{2}m\omega^2x^2$, the wavefunction of ground state is given as $\phi_0 = \left( \frac{m\omega }{\pi \hbar} \right)^{\frac{1}{4}} exp \left( -\frac{m\omega}{2\hbar} x^2 \right)$.

Now an external electric field $E$ is applied.

Part (a): Find the new energies and wavefunction of the ground state.

Part (b): Find the probability that the particle will be in the ground state of the new potential.

Homework Equations

The Attempt at a Solution

The Hamiltonian now becomes:

$$H = \frac{p^2}{2m} + \frac{1}{2} m\omega^2 \left(x - \frac{qE}{m\omega^2} \right)^2 - \frac{q^2E^2}{2m\omega^2}$$

Thus the shift in energy is $\frac{q^2E^2}{2m\omega^2}$. New energies are given by: $E_n = (n+1)\hbar \omega - \frac{q^2E^2}{2m\omega^2}$.

This represents a displaced harmonic oscillator, with eigenfunction:

$$\phi_0 \space ' = \left( \frac{m\omega}{\hbar \pi} \right)^{\frac{1}{4}} e^{-\alpha (x-\frac{qE}{m\omega^2})^2 }$$

Part(b)

Do I overlap this state with the old one and integrate?

 

[bloby]

Edit: yes
Edit take two: may I suggest to edit for typos not to change completely the questions?