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1. The problem statement, all variables and given/known data

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

[tex] V(x) = 0.5 * m \omega^2 (x - \frac{qE}{mw^2})^2 - \frac{1}{2m}(\frac{qE}{\omega})^2 [/tex]

I know the solutions to the TISE:

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

(Those are different Es)

So, I plug V(x) into the TISE and get:

[tex] -\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) [/tex]

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?

2. Relevant equations

3. The attempt at a solution

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# QM simple harmonic oscillator

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