Understanding the Infinite Well Potential for Modeling Electron Bound to Atom

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Discussion Overview

The discussion revolves around the modeling of an electron bound to an atom using the infinite well potential. Participants explore the implications of having a potential of zero within the well and its relation to the concept of a bound particle versus a free particle.

Discussion Character

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

Main Points Raised

  • One participant questions the physical meaning of having a potential V(x) = 0 for all x in the context of modeling a bound electron.
  • Another participant suggests that if V(x) = 0 for all x, it describes a completely free particle, indicating a misunderstanding of the original question.
  • A participant expresses confusion about how a zero potential inside the well can represent a bound particle and inquires about restrictions on the quantum number n for free particles.
  • One response clarifies that an electron in a box with impenetrable walls is akin to being in an infinite potential well, where V = 0 inside the well, confining the electron to a finite region of space.
  • Another participant argues that the infinite square well does not model any physical scenario accurately, noting its role as a simplified test case in quantum mechanics to illustrate energy level quantization.
  • A later reply proposes using the Schrödinger equation in spherical coordinates to better model the electron's attraction to the nucleus, suggesting a potential that varies with distance rather than being constant or infinite.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of the infinite well potential to real physical systems, with some arguing it serves as a useful simplification while others contend it lacks physical relevance. The discussion remains unresolved regarding the best model for an electron bound to an atom.

Contextual Notes

There are limitations in the assumptions made about the potential energy and its implications for modeling bound versus free particles. The discussion also touches on the mathematical representation of potentials, which may depend on specific conditions or definitions.

swain1
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I am just trying to get my head round how this models the electron bound to an atom. I don't understand why the potential is zero in the well What physical case corresponds to the condition that V(x)=0 for all values of x?
Thanks
 
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If V(x) = 0 for all x (as opposed to only inside the well), then you have a completely free particle, with no net force acting on it. Is that what you were after, or did I misunderstand your question?
 
Yes it was, that is what I thought it would be but then I was wondering why the potential could be zero inside the well as this is meant to represent a bound particle.
Also for a completely free particle, would there be a restriction on the value of n? cheers
 
If the electron is in a box with impenetrable walls, then it's equivalent to being in an infinite potential well, in this case with V=0 inside. That is, the problem describes an electron confined to a finite region of space with the only forces acting during collisions with the walls.

Regards,
Reilly Atkinson
 
The infinite square well doesn't really model anything physical. The closest thing that it comes to modeling is a finite quantum well used in semiconductor lasers. However, the square well is basically the simplest test case that you can construct in QM, since it illustrates the quantization of energy levels.
 
What u might be looking for is the schrödinger equation expressed in radius and angle. You can then make a much more accurate picture as you can use the attraction of the electron to the nucleus as the potenital in the from U(x)= -ke^2 /r. This gives a much more accurate picture of an electron round an atom, as the potential isn't infinite or 0, but increases with distance. Hope this helps.
 

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