Is there any time-dependent hydrogen atom Schrödinger equation, solvable analytically?

Question

It's well-known that hydrogen atom described by time-independent Schrödinger equation (neglecting any relativistic effects) is completely solvable analytically.

But are any initial value problems for time-dependent Schrödinger equation for hydrogen solvable analytically - maybe with infinite nuclear mass approximation, if it simplifies anything? For example, an evolution of some electron wave packet in nuclear electrostatic field.

Answer 1

Of course there are. That's why we solve the time-independent version of the Schrödinger equation to begin with: because given any eigenfunction $\psi_0$ of the hamiltonian with eigenvalue $E$, the phase-evolved combination $$ \psi(t) = e^{-iEt/\hbar}\psi_0 $$ is a solution of the time-dependent Schrödinger equation, and, moreover, any linear combination of such solutions is still a solution.

There is, of course, an understandable prejudice against taking a stationary state as an initial condition for the TDSE (but it is just a human prejudice with no real meat to back it up). If that really bothers you, then you can just take a nontrivial linear combination, like, say, $$ \psi = \frac{\psi_{100}+\psi_{210}}{\sqrt{2}}, $$ and it will then show oscillations in both the position-space and momentum-space probability distributions. To borrow from my answer to Is there oscillating charge in a hydrogen atom?, the explicit wavefunction is given by

$$ \psi(\mathbf r,t) = \frac{\psi_{100}(\mathbf r,t) + \psi_{210}(\mathbf r,t)}{\sqrt{2}} = \frac{1}{\sqrt{2\pi a_0^3}} e^{-iE_{100}t/\hbar} \left( e^{-r/a_0} + e^{-i\omega t} \frac{z}{a_0} \frac{ e^{-r/2a_0} }{ 4\sqrt{2} } \right) , $$ and this goes directly into the oscillating density: $$ |\psi(\mathbf r,t)|^2 = \frac{1}{2\pi a_0^3} \left[ e^{-2r/a_0} + \frac{z^2}{a_0^2} \frac{ e^{-r/a_0} }{ 32 } + z \cos(\omega t) \, \frac{e^{-3r/2a_0}}{2\sqrt{2}a_0} \right] . $$

Taking a slice through the $x,z$ plane, this density looks as follows:

This combination gives you an explicit analytical solution of the time-dependent Schrödinger equation. Now, again, it is understandable to dismiss this as somehow "not being a real wavepacket", partly because from some perspectives it might feel "too easy", but all of those are human prejudices with very little support on well-defined and truly meaningful mathematical criteria on the initial conditions or the corresponding solution. This is an honest-to-goodness electronic wavepacket moving under the influence of a point-charge nucleus.

Answer 2

What you do have available is an explicit knowledge of the eigenvalues and eigenvectors (also for the continuous spectrum). By expanding your initial wavepacket in terms of the eigenvectors you then obtain its value for later times as a sum (or integral for continuous spectrum) with added weight factors exp[-i$\lambda$t], where $\lambda$ is the eigenvalue associated with the corresponding eigenvector.

Answer 3

Solution of an initial value problem can be written as integral of the initial function $\psi_0$ multiplied by the propagator of the Schr. equation. Depending on the function $\psi_0$, the integral may or may not be calculable in terms of simple functions. I do not know of any initial function $\psi_0$ and potential $A(t)$ that would admit simple exact solution; the equation with time-dependent term is difficult to solve. More rewarding way seems to be to find the solution with a computer. The real problem is I think elsewhere - how do we find appropriate function $\psi_0$ to describe real atoms? Often the first eigenfunction of the Hamiltonian is used, but I do not think this is particularly well motivated.