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Quantum translational motions can be modeled with the particle in a box model and rotation and vibration can be modeled and harmonic oscillator models, respectively. Is the square potential only modelled for an electron in an atomic bound state, and harmonic oscillator for molecular states? Or they can both be used for each. Is the particle in a box only valid for one particle, and the harmonic oscillator potential for two particles?

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The square potential is mostly a simple-to-solve toy model used to gain intuition. Two of the main things you are supposed to take out of the square well potential are the difference between scattering states and bound states, and the existence of quantum tunneling. The square well also serves as an entry point for condensed matter systems, since a periodic lattice of square wells is a simple toy model of a crystal and lets you understand band structure.

The square well has some limited uses in understanding real systems, for example it can approximately describe quantum dot and quantum well systems.

The harmonic oscillator, on the other hand, is an enormously useful system. It's the workhorse of theoretical physics. Essentially any stable system can be described as a harmonic oscillator, for small oscillations. You gave one example application of vibrational and rotational molecular states. But the harmonic oscillator is also used in particle physics to describe particle states; it is used in quantum optics to describe modes of the electromagnetic field; it is used in condensed matter to describe electrons bound to nuclei and to describe phonon modes... The harmonic oscillator is a tool that needs to be in the toolbox of any working theoretical physicist.

Furthermore, the ladder operator method you learn in solving the harmonic oscillator problem is incredibly important and useful and has many applications that have nothing to do with the harmonic oscillator.

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  • $\begingroup$ Thank you. So does it can use the harmonic oscillator potential to model an electron state function in Bohr's atom? $\endgroup$
    – LION
    Commented Feb 15, 2021 at 4:34
  • $\begingroup$ @LION That's not the best model for atomic structure. Usually the starting point is to solve the Schrodinger equation with a Coulomb potential. Additional physics (relativistic corrections, spin-orbit couplings, etc) can be added to this using perturbation theory. Wikipedia is a decent starting point for this: en.wikipedia.org/wiki/Hydrogen_atom#Schr%C3%B6dinger_equation $\endgroup$
    – Andrew
    Commented Feb 15, 2021 at 12:38
  • $\begingroup$ Thank you. The energy level spacings in the harmonic oscillator potential are equally spaced, that is different from the binding energy levels around the atom. So, I wondered if it could be a valid model. $\endgroup$
    – LION
    Commented Feb 15, 2021 at 23:49
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    $\begingroup$ @LION Right, it's a reasonable thought, but the $1/r$ potential has no location where you can perform a Taylor expansion and find the potential has the form of a harmonic oscillator, $V(x)={\rm const}+\frac{1}{2}V''(x_0)(x-x_0)^2+\mathcal{O}((x-x_0)^3)$. There is always a linear term, or else the singularity at $r=0$. $\endgroup$
    – Andrew
    Commented Feb 16, 2021 at 0:04
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In addition to the link on applications of the harmonic oscillator provided in the comments to your question, note that the finite well has important applications in nuclear physics as the Woods-Saxon potential is reasonably well approximated by a finite square well in 3d. It is then possible to obtain some crude understanding of energy levels splitting with or without angular momentum without resorting to numerical methods: the solution here are in terms of spherical Bessel functions.

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Likewise, various double-wells (v.g. for the ammonia molecule) can be modelled by an infinite well with a bump in the middle. (Other applications include quantum wells.)

Overall, the square well - finite or infinite - is useful because solvable with minimum technology (even if, in the finite case, one must find intersections of curves numerically).

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