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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.

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What do you mean by "analytically"? You probably don't mean the math definition, which is that the function converges to its Taylor series. If you mean "involving simple functions" the you should know there's no qualitative difference between numerical integration and special functions. In fact many common special functions are evaluated by your computer via the differential equation they satisfy. –  Chris White Feb 23 at 22:44
    
@ChrisWhite I mean explicit solution, in terms of such functions, which don't require to set up a dense spatial grid and propagate the solution in small temporal steps to find the value at a given point in spacetime with required precision. –  Ruslan Feb 24 at 4:15

3 Answers 3

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.

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The problem with this approach is that one needs to do lots of numeric 3D integrations to find projections of initial state on eigenstates. And if the initial state has too high localization so that much of continuous spectrum states have contribution to initial state, then one would have to do even more integrals. This approach doesn't seem as a very good one in general, this is why I asked about direct solution of time-dependent equation. –  Ruslan Jan 28 at 12:34
    
Yes, I see your problem. Did you consider a Green's function approach? –  Urgje Jan 29 at 11:31
    
I don't know much about Green's functions. Could you elaborate on how to apply them to this problem (and what to read to understand your explanation better)? –  Ruslan Jan 29 at 11:38
    
The Green's function of the time-independent case is known, both in coordinate and momentum representation. Whether or not that helps in your case I do not know. –  Urgje Jan 30 at 16:08

Urgje gave you the answer. In its basic form (Schrödinger), the Hamiltonian is time-independent, therefore the general theory will tell you how to write the general solution of the Schrödinger equation as the sum/integral of the solutions of the spectral equation weighed by time-dependent exponentials.

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Why repeat already existing answer, especially if comments under it say why it's not acceptable? –  Ruslan Feb 23 at 16:58

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.

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My question was "are there any initial value problems", not "what is the definite solution". Of course, by initial conditions I mean not such trivial ones as superposition of finite eigenfunctions, but some form of wave packets. –  Ruslan Mar 25 at 19:06
    
OK, I've edited my answer accordingly. Sorry I can't be of more help. –  Ján Lalinský Mar 26 at 21:07

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