# QM simple harmonic oscillator

by ehrenfest
Tags: harmonic, oscillator, simple, solved
 P: 1,996 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: $$V(x) = 0.5 * m \omega^2 (x - \frac{qE}{mw^2})^2 - \frac{1}{2m}(\frac{qE}{\omega})^2$$ 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? 2. Relevant equations 3. The attempt at a solution
 Emeritus Sci Advisor PF Gold P: 11,155 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.