Time dependent perturbation from Galilean transformation - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-07-17T20:29:39Z https://physics.stackexchange.com/feeds/question/329516 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://physics.stackexchange.com/q/329516 1 Time dependent perturbation from Galilean transformation John Page https://physics.stackexchange.com/users/149140 2017-04-28T03:57:13Z 2017-04-30T21:55:40Z <p>I'm having a little trouble figuring out how to start this. The question is as follows. At time $t&lt;0$ a hydrogen atom is in the rest frame $\mathfrak{R}$. Then at time $t=0$ the atom suddenly starts moving at a speed $V$ which is non-relativistic. It is now described in the rest frame $\mathfrak{R}'$ under the transformation $$t' = t, \, \, \, \vec{r}' = \vec{r}-\vec{V}t$$ The question asks to find the probability that at time $t&gt;0$ in rest frame $\mathfrak{R}'$ that the electron in the hydrogen atom is still in it's ground state.</p> <p>I know so far that this is a time dependent perturbation theory. I also know the probability can be computed by using $$|C_m^{(1)}(t)|^2$$ where $$C_m^{(1)}(t) = -\frac{i}{\hbar}\int_0^t dt e^{i\omega_{m,i}t}H'_{m,i}(t)$$ and $$\omega_{m,i}= \frac{E_m-E_i}{\hbar}, \, \, \, H'_{m,i} = \langle\psi_m^0|\hat{H}'(t)|\psi_i^0\rangle$$ What I'm not clear on is what $E_m$, $\psi_m^0$ and $\hat{H}'(t)$ should be. Would these just be the Galilean transformations of $E_i$, $\psi_i^0$ and $\hat{H}(t)$ for a hydrogen atom, or is there something more subtle I'm missing here?</p>