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

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Discussion Overview

The discussion revolves around the characteristics of a potential well that is proportional to -1/|x|, specifically addressing the nature of energy levels within this potential. Participants explore whether the number of possible energy levels is finite or infinite, the implications of a horizontal asymptote, and the potential well's applicability to the hydrogen atom.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions whether the potential well allows for a finite or infinite number of energy levels, suggesting that the infinite width before the asymptote implies infinite levels.
  • Another participant asserts that the potential well can describe the hydrogen atom in three dimensions, noting the importance of using |r| instead of |x|.
  • A different participant clarifies that while a one-dimensional model can be useful, the properties of energy levels differ from the three-dimensional case, particularly regarding the spacing of energy levels as one approaches higher excited states.
  • One participant expresses confusion about the relationship between bound and unbound states, asking if there is a maximum energy level (13.6 eV) for bound states and whether unbound states are infinite above this level.
  • Another participant explains that the ground state energy is -13.6 eV and distinguishes between the continuum of free states (E ≥ 0) and discrete bound states (E < 0).

Areas of Agreement / Disagreement

Participants express differing views on the nature of energy levels in the potential well, with some suggesting infinite levels and others pointing to a maximum energy level for bound states. The discussion remains unresolved regarding the implications of the horizontal asymptote and the relationship between bound and unbound states.

Contextual Notes

There is uncertainty regarding the definitions of bound and unbound states, as well as the implications of the potential well's shape on energy levels. The discussion does not resolve these complexities.

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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?
 
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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.
 
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.
 
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?
 
I think this picture may be helpful:

image24827.gif


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.
 

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