Time-dependent harmonic oscillator I am dealing with the problem to solve the following Schroedinger equation: 
$$i\hslash\partial_{t} \Psi = ( -\nabla^2 +w^2(t) )\Psi$$
where the frequency of the oscillator depends on time. I tried to resolve it by using the Kruskal approach, as described in the following paper 

"An Exact Quantum Theory of the TimeDependent Harmonic Oscillator and
  of a Charged Particle in a TimeDependent Electromagnetic Field",
  H. R. Lewis and W. B. Riesenfeld, 

but without any results. 
In particular I am having problems to find the $\rho$ function. 
In my case the frequency $w^2(t)$ is: 
$$e^{4(t+Ce^{t})}$$ where $C$ is a generic constant. Can you help me? 
 A: (No mass term and no factors of 1/2?  It is weird to me that you included $\hbar$ and not these other factors)
It seems like you may be able to use separation of variables.  Just assume (I am going to just use 1 dimension right now as it should be the same process for 3)
$$\Psi = \psi(t)\phi(x)$$ 
This gives:
$$i \hbar \;\partial_t \psi (t) \phi(x) = -\hbar^2\psi(t)\partial_x^2\phi(x)+\omega^2(t)\psi(t)\phi(x)$$
Now divide by $\psi(t)\phi(x)$
$$ i \hbar \;\frac{\partial_t \psi (t)}{\psi(t)} = -\hbar^2\frac{\partial_x^2\phi(x)}{\phi(x)}+\omega^2(t) $$
Now assume $\frac{\phi_x^2 \psi (t)}{\phi(x)}$ is equal to a constant $K$ and you get two differential equations:
$$-i\hbar \frac{\partial_t \psi (t)}{\psi(t)} = \hbar^2 K + \omega^2(t)$$
$$\partial_x^2\phi(x) = K \phi(x)$$
The second one (the spatial one) is easily solved to be a sum of exponentials and the first one (I believe) can also be solved, perhaps using an integrating factor since it is only first order.  You probably won't be able to put it in closed form, but it will be an integral solution that can be numerically calculated or put into an expansion form.
I haven't looked at the details of this but I think what I say will work....
