# Oscillator solutions with regard to TDSE Eigenvalues

Taking a shot at a QM harmonic oscillator problem tonight.

Consider a 1-D harmonic potential: $$V(x) = \frac {m\omega^{2}x^{2}} {2}$$ solve for the gen. solution to the TDSE, $\psi(x,t)$ utilizing HO eigenstates, $\phi_{n}(x)$

Maybe if it was time 'independent', yes. I could easily draft up a solution for the TISE just by inserting $\psi(x) = e^{\frac{x^2}{2b^2}}$ into $$-\frac {\hbar} {2m} \frac {d^{2}\psi(x)} {dx^{2}}+\frac{1} {2}m\omega^{2}x^{2}\psi(x)=E\psi(x)$$ to get $$\frac {d^{2}\psi(x)} {dx^{2}}+\frac {1} {b^{2}}\psi(x)-\frac {x^{2}} {b^4}\psi(x) = 0$$.

From there on out it really isn't that difficult save a few substitutions. But the time dependence throws me. Something I'm missing?

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