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In quantum mechanics we have different models like the infinite and finite square well, potential barrier, etc. What I don't understand is how these are applicable to real life situations. For example, in a one dimensional potential well we have two points where the potential goes to infinity. Now, if we were to apply this in real life situations then what would these "potentials" be? Can we take the example of two parallel plates with large voltage and a single particle between them? If yes, then in what other ways can this model be used? Also, does the potential used in the model refer only to the electrostatic potential or can it mean something else in other situations.

Edit: I am not looking for only the one dimensional potential well and took the example here just to explain my doubts. I just want to know how real life situations can be solved with quantum modelling. Also, what does negative energy in a quantum system mean?

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  • $\begingroup$ From Greiner's QM book about infinite 3D well: "For example, in nuclear physics all nucleons in a nucleus are supposed to be in a potential well. Of course this potential well is spherically symmetric, but for small nuclei a boxlike potential is an acceptable approximation." $\endgroup$
    – jw_
    Commented Feb 1, 2020 at 2:49

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These models may seems really strange when you learn QM for the first time since they don't have much sense in real problems.

In QM books, these examples are mainly used to demostrate the process of applying Schrodinger's equation to some problems. And we alwasy start with the easiest one, which is one-dimensional infinite potential, then one-dimensional finite potential, then tree-dimensional.

The first very exact real problem in the book will be the hydrogen atom, which will be far more complex to solve than the above problems. Everything suddenly make sense to the real world when this come out.

The above simple models has some application to real problems, too, like any problem in which the particle is restricted to a finite region, but in most case the result is very rough since too much approximations are made.

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Here is a real-wordish application of the one-dimensional well. Consider the problem of non-interacting particles in a three-dimensional box. The three spatial directions decouple in this situation so that one may consider three one-dimensional problems instead. A real box will have some permeability, but let's assume it to be negligable. This can be modelled by letting the potential rise infinitely steep.

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  • $\begingroup$ Like the electrons in a metal cube that are confined by the work function. $\endgroup$
    – user137289
    Commented Oct 13, 2019 at 8:35
  • $\begingroup$ @pieter, yes that is a valid application. $\endgroup$
    – Lorenz Mayer
    Commented Oct 13, 2019 at 8:38
  • $\begingroup$ @Pieter so the "potential" in this case is the work function of the material. And since we can overcome the potential by imparting energy as in the photoelectric effect then this electrons in a metal cube problem can be modelled by the finite potential well. Am I understanding it correctly? $\endgroup$
    – Korra
    Commented Oct 13, 2019 at 8:52
  • $\begingroup$ @Korra There are few cases where it matters that the surface barrier is finite, an infinite barrier usually explains the phenomena. $\endgroup$
    – user137289
    Commented Oct 13, 2019 at 9:02
  • $\begingroup$ @Pieter but if the barrier is infinite then it wouldn't really explain photoelectric effect. So in that case we have to use the finite barrier right? $\endgroup$
    – Korra
    Commented Oct 13, 2019 at 9:05
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The energies of sigma bonding and antibonding orbitals of diatomic molecules can be estimated by the eigenvalues of an electron of a one-dimensional box with a length three times as large as the nuclear separation.

This works nicely especially for the unbound resonances. I can look up the reference if you want it.

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  • $\begingroup$ So using the one dimensional potential well we can find the energy of the electron that is participating in the bonding? I don't understand why we are taking length as three times larger than the nuclear separation. $\endgroup$
    – Korra
    Commented Oct 13, 2019 at 8:54
  • $\begingroup$ @Korra The nuclear potential wells are only deep very close to the molecular axes. A bit a way from the molecular axis (which represents a much larger volume), the wave functions are approximately free. The electrons are confined to the total length of the molecular potential well which is about three times the internuclear distance. $\endgroup$
    – user137289
    Commented Oct 13, 2019 at 9:15

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