# Solutions of the Harmonic Oscillator are $not$ always a Combination of Separable Solutions?

Are there solutions of the Schrödinger equation that are not a linear combination of separable solutions and how do we find them?

In Griffiths, Quantum, Prob. 2.49, there is a solution of the (time-dependent) Schrödinger equation, which reads $$\Psi(x,t)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\left[-\frac{m\omega}{2\hbar}\left(x^2+\frac{a^2}{2}(1+e^{-2i\omega t})+\frac{i\hbar t}{m}-2axe^{-i\omega t} \right)\right].$$ It seems that this is not a linear combination of the stationary states that he found previously in the chapter.

If it is the caes, does that mean that solving the time-dependent Schrödinger equation by separation of variables does not yield the general solution as the author claimed? if so, how do we find the other solutions?

• Why do you say that "it is clear that this is not a linear combination of the stationary states that he found previously in the chapter"? It is not at all clear to me! Have you looked at Mehler's formula? (see Wikipedia for this) Sep 28 '20 at 12:12
• @mikestone I should have said it seems not a linear combination of the stationary states, thanks for this note!, I'll edit it Sep 28 '20 at 12:15
• @mikestone should I remove the question ? Sep 28 '20 at 12:18
• No! I just posted an "answer" of a sort. Sep 28 '20 at 12:29

Sometimes the expansions are not obvious. For example The harmonic oscillator time-dependent Schr"odinger equation $$i\partial_t \psi = -\frac 12 \partial^2_x \psi +\frac 12 \omega^2 x^2 \psi$$ has a `breathing'' solution $$\psi(x,t)= \left(\frac{\omega}{\pi}\right)^{1/4}\frac 1{\sqrt{e^{i \omega t} +R e^{-i\omega t}}}\exp\left\{ - \frac \omega 2 \left(\frac{1-R\,e^{-2i\omega t}}{1+R\,e^{-2i\omega t}}\right)x^2\right\},$$ where the parameter $$|R|<1$$.
$$\psi(x,t) {=}\pi^{1/4}\sum_{n=0}^\infty e^{-i(n+1/2) \omega t} \varphi_n(0)(i\sqrt R)^n \frac{\varphi_n(\sqrt{\omega} x)}{(\omega)^{1/4}}.$$ Here $$\varphi_n(x)\equiv \frac{1}{\sqrt{2^n n! \sqrt{\pi}}} H_n(x) e^{-x^2/2}$$ is the normalized $$\omega=1$$ harmonic oscillator wavefunction. Now $$\varphi_n(0)$$ vanishes if $$n$$ is odd, and $$\pi^{1/4}\varphi_{2n}(0)= \frac{1}{\sqrt{4^n (2n)! } } \frac{(2n)!}{n!}(-1)^{n}.$$ so one has found as set of quite "non obvious" expansion coefficients.