Characteristics of a Potential Well that is Proportional to -1/|x|

[WoahWoah]

Regarding a potential well that is proportional to -1/|x|, are the amount of possible energy levels finite or infinite? (The potential well is narrow in the middle and approaches a horizontal asymptote as you leave the middle, like the shape of a tornado).

I figured it would be infinite, because the well gets infinitely wide before the horizontal asymptote so that energy levels of any "length" could fit between the walls. But It doesn't really make sense if there's a finite max potential energy (~horizontal asymptote) and the professor said there was a flaw in my reasoning.

I thought about it for a while and couldn't seem to find the explanation.

Lastly, two quick questions: is it correct to say that as you approach the horizontal asymptote, the energy levels get infinitely close together? I'm guessing it's wrong because that implies an infinite amount of energy levels. And can the -1/|x| potential well be used to describe the hydrogen atom?

 

[tom.stoer]

This potential describes the hydrogen atom perfectly well, but of course in three dimensions, so it should read |r|, not |x|. Every book of lectures on QM discusses this in detail.

 

[hilbert2]

You are describing a one-dimensional hydrogen atom, which is a useful model system in some applications, as far as I know. The properties of the solutions (eigenfunctions/values) differ from the 3D case. See http://arxiv.org/ftp/quant-ph/papers/0608/0608038.pdf .

The spacing between energy levels really gets arbitrarily small as one goes to higher and higher excited states. Also, the total energy of the system is NOT bounded from above, there exists a continuous spectrum of unbound scattering states above the discrete set of bound states.

 

[WoahWoah]

Thanks for the replies. Sorry I'm completely lost when it comes to physics, so my understanding now is that the spacing between energy levels get smaller until a point where the electron ionizes. I'm guessing that point is at 13.6eV? Sorry if I still don't understand, does this mean that although there are an infinite amount of bound states, there is still a max energy level (13.6eV)? And above that max energy level, there is an infinite amount of unbounded states?

 

[tom.stoer]

I think this picture may be helpful:

The continuum of free states is the range E ≥ 0.
The discrete bound states are in the range E < 0.
Ionization means to bring an electron from E < 0 to E > 0.
The ground state has an energy of E = -13.6 eV.