Shaken harmonic oscillator I cannot find in literature an exact expression of the probability of transition of the eigenvectors of the quantum harmonic oscillator Hamiltonian when it is "shaken" in this way:
\begin{equation} H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2(x-x(t))^2\end{equation}
I found only first order approximations with a "forcing" term $m\omega^2 x(t)x=F(t)x$, which is well known in literature, but this is not enough for me, since I am looking for high (>0.5) transition probabilities, not reachable with the first order approximation.
 A: Let's look for a solution to the nonstationary Schrodinger equation in this problem in the following form:
$$
|\Psi,t\rangle = \hat{U}(t)|\Psi,t\rangle_0\quad\mbox{where}\quad \hat{U}(t) = e^{-\frac{i}\hbar S(t)} e^{-\frac{i}\hbar\hat{p}x_1(t)} e^{\frac{i}\hbar\hat{x}p_1(t)}
$$
It is possible to choose functions $x_1(t)$, $p_1(t)$ and $S(t)$ in such a way that $|\Psi,t\rangle_0$ would be a solution to the Schrodinger equation for the unperturbed oscillator:
$$
i\hbar\frac{\partial}{\partial t}|\Psi,t\rangle_0 = \hat{H}_0|\Psi,t\rangle_0\quad \mbox{where}\quad \hat{H}_0 = \frac{\hat{p}^2}{2m} + \frac{m\omega^2}2 \hat{x}^2.
$$
We just need to satisfy the following equality
$$
i\hbar \hat{U}^{-1}(t)\frac{\partial}{\partial t}\hat{U}(t) = \hat{U}^{-1}(t)\hat{H}(t)\hat{U}(t) - \hat{H}_0.
$$
The operator $\hat{U}(t)$ properties are
$$
\hat{U}^{-1}(t)\hat{x}\hat{U}(t) = \hat{x} + x_1(t),\quad \hat{U}^{-1}(t)\hat{p}\hat{U}(t) = \hat{p} + p_1(t),
$$
$$
\hat{U}^{-1}(t)i\hbar\frac{\partial}{\partial t}\hat{U}(t) = \frac{\partial S}{\partial t}(t) + \dot{x}_1(t)p_1(t) + \dot{x}_1(t)\hat{p} - \dot{p}_1(t)\hat{x}
$$
Hence, functions $x_1(t)$, $p_1(t)$ and $S(t)$ should be solution to the following system of equations
$$
\dot{x}_1(t) = \frac{p_1(t)}m,\quad \dot{p}_1(t) = - m\omega^2(x_1(t)-x(t)),
$$
$$
\frac{\partial S}{\partial t}(t) = -p_1(t)\dot{x}_1(t) + \frac{p_1^2(t)}{2m} + \frac{m\omega^2}2(x_1(t)-x(t))^2.
$$
I think the last problem has a well-known solution. Now it is up to you to find this solution and to express the transition probability.
