Solution to Schroedinger Equation for a huge hypothetical

[Sven Andersson]

Solution to Schroedinger Equation for a huge hypothetical atom?

Let's say you have hypothetical atomic nucleus with a very large Z, say a million times of the most highly charged ones; what would be the solution to the SE at very large n's i.e. at very large distances, say several centimeters, from the hypothetical nucleus?

S.A.

 

[Simon Bridge]

Welcome to PF;
... short answer: complicated.
Depends what you want to find out... the basic prediction is that the nucleus will break apart in a very short time, and that the electron binding energies would be, in any case, very small.

Note: It is very difficult to find exact solutions to the Schrödinger equation for systems of more than one or two particles.

 

[Sven Andersson]

Clarification; I meant a solution to SE for a two body problem; one electron and one nucleus with Z=1 000 000. Let's assume that the nucleus is stable and has atomic weight 2 000 000. And the solution I want is for n=1000 or higher (at a distance of several centimeters). What values do you plug in? What does the equation look like?

 

[Nugatory]

So you're looking for the solution for an electron bound in the field of a fixed point charge... That two body problem is just the hydrogen atom with potential $U(r)=\frac{-Ze}{4\pi\epsilon_0{r}}$

Google for "hydrogen atom Schrödinger" will find it.
It won't be even slightly realistic for many reasons, but that's a different problem.

 

[TeethWhitener]

I meant a solution to SE for a two body problem; one electron and one nucleus with Z=1 000 000. Let's assume that the nucleus is stable and has atomic weight 2 000 000.

I suggest once you find the eigenfunctions to this problem, you do a quick evaluation of the expected value for the momentum of an electron in the ground state. Then divide by electron mass to get the average speed of the electron. Then come back and tell me why such a system can't exist (or is overlooking something fundamental).

 

[Simon Bridge]

What they said... also, the distance from the nucleus is a trickier concept than how you appear to be thinking.
Such conceptual exercizes can be good for uncovering these nuances.