[plus]

It's possible to simulate the time-evolution of the electron density using the time-dependent Schrodinger equation. That said, if the initial state is chosen to be an eigenfunction of the Hamiltonian (read: any of the things being plotted on this page), there will be no time dependence -- the electron density will remain static. However, if the initial state is chosen as a superposition of states (read: any configuration that is not given by the square amplitude of an eigenfunction of the Hamiltonian), you can simulate the time evolution.

[ellis-bell]

that's not entirely true though. you're right about the superposition of energy eigenstates, and what i'm about to say is uninteresting from the perspective of the posted simulation, although it is interesting "in the real world"

in time dependent perturbation theory you can show that electrons can transition to different energy states through spontaneous emission. for example, psi(n=4,l=0)-->psi(n=3, l=1) by emitting a photon.

thus, the "change" in the simulation posted here would be uninteresting since it would just correspond to clicking a different eigenstate in the top right!

[plus]

What I'm saying is true -- if the initial state is chosen to be a superposition of states, then the time-dependent Schrodinger equation enables one to simulate the time evolution of the electron density.

> thus, the "change" in the simulation posted here would be uninteresting since it would just correspond to clicking a different eigenstate in the top right!

That is a drastic oversimplification of reality. Spontaneous emission of a photon is not instantaneous. It is still possible to simulate the time-evolution of this process.

[ellis-bell]

everything is a simplification. of course it's not spontaneous, but that is literally the word used by physicists to describe the phenomenon.

what does the wave equation of the electron in a hydrogen atom look like during "spontaneous" emission of a photon? i don't think anyone has any idea.

i'm talking about something entirely separate from a linear combination of two energy eigenstates. i'm not saying take

\psi = \sin{\theta} \psi_1 + \cos{\theta} \psi_2

where \psi_1 and \psi_2 are eigenstates of the hydrogen atom hamiltonian and simulate it. i'm saying there is a phenomenon that i'm pretty sure wouldn't be adequately modelled by a smooth function.

edit: add explanation of \psi_1 and \psi_2

[plus]

The hydrogenic orbitals form a complete and orthonormal basis in which you can expand any arbitrarily-chosen one-electron wavefunction. That is to say that any arbitrarily chosen initial state for the one-electron wavefunction (and thus any arbitrarily chosen electron density) can be expressed as a linear combination of the hydrogenic orbitals. This, in combination with the time-dependent Schrodinger equation, enables us to simulate the time evolution of the hydrogenic system starting from any arbitrarily chosen initial state.

> what does the wave equation of the electron in a hydrogen atom look like during "spontaneous" emission of a photon? i don't think anyone has any idea.

It sounds like you are not aware that this is an entire field of research within the theoretical chemistry community. Theoreticians have been studying spectroscopy for a century.

[vertbhrtn]

Would it really just jump into another shape? I don't understand QM much, but the Schrodinger equation looks like a typical heat equation (with some complex-number trickery) and thus in general should evolve smoothly. There should be a transition period between two shapes.

[plus]

You are spot on here: the heat equation and the Schrodinger equation are both special cases of the wave equation. A major difference between the two is that the Schrodinger equation deals with complex numbers, as you state. Transition between states is not instantaneous, though it is very fast, on the scale of attoseconds (10^-18 seconds).

[vertbhrtn]

What if something else chosen as an initial condition, like a sphere or some random shape? Would it converge to something meaningful if we apply the time-dependent SE with a meaningful field potential?

[plus]

Without some way of releasing or absorbing energy, the system would oscillate forever.