How Does an Electric Field Affect the Quantum Harmonic Oscillator?

[ehrenfest]

[SOLVED] QM simple harmonic oscillator

Homework Statement

If I have a particle in an SHO potential and an electric field, I can represent its potential as:

$$V(x) = 0.5 * m \omega^2 (x - \frac{qE}{mw^2})^2 - \frac{1}{2m}(\frac{qE}{\omega})^2$$

I know the solutions to the TISE:

$$-\hbar^2 /2m \frac{d^2 \psi}{ dx^2} + 0.5 m\omege^2 x^2\psi(x) = E\psi(x)$$ (*)

(Those are different Es)So, I plug V(x) into the TISE and get:

$$-\hbar^2 /2m \frac{d^2 \psi}{ dx^2} + (0.5 * m \omega^2 (x - \frac{qE}{mw^2})^2 - \frac{1}{2m}(\frac{qE}{\omega})^2) \psi(x) = E\psi(x)$$Now, since we only shift and translated the potential, I should be able to find a substitution for x that yields the equation (*) in a new variable y = f(x), right?
The problem is, after I move the constant term to the RHS, I cannot find the right substitution. What am I doing wrong?

Homework Equations

The Attempt at a Solution

 

[ehrenfest]

I think that I can even prove that there is no constant that you can add to x to find a suitable substitution. Something must be wrong here?

 

[Gokul43201]

Have you tried the obvious substitution: $\xi = x-qE/m\omega ^2 ~$ ?

You can ignore the additive constant and refer all energies relative to that value.

 

[ehrenfest]

Yes, I figured it out. The problem was that I was under the false impression that I had to substitute epsilon for every x in that equation, which does not work.

I realized, however, that you can substitute for psi(x) separately since it is a factor on both sides of the equation.