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I was thinking about the relationship of classical mechanics to quantum mechanics, as I just took my first course in quantum mechanics. My specific question was about quantization. For a harmonic oscillator, the way to "quantize" it is to take classical observables to operators, and using the Hamiltonian operator to solve for the wavefunction using Schrodinger's equation.

The problem I am having is I'm not clear on interpretation. Are we quantizing a classical harmonic oscillator to get an idea of what a quantum harmonic oscillator would behave like? Or, are we quantizing a classical harmonic oscillator to see how a classical harmonic oscillator works at microscopic/quantum scales?

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    $\begingroup$ I am not sure I understand the difference between your two final questions. Also small note, the quantization you refer to comes from finding the (energy) eigenvalues of the Hamiltonian. You don't necessarily even have to "solve for the wavefunction (technically eigenfunctions)" $\endgroup$ – Aaron Stevens Apr 2 '19 at 18:23
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While "quantizing" a classical theory (such as that of a harmonic oscillator), what we do is that we try to find (in particular, guess) a consistent quantum theory whose (one) classical limit would be the classical theory with which we started out.

I don't think there is a distinction between "a quantum harmonic oscillator" and "a classical harmonic oscillator at a quantum scale". In particular, because, the only thing one can possibly mean with "a classical harmonic oscillator at a quantum scale" would be a quantum theory which looks like the classical harmonic oscillator in (one of) its classical limit(s)--and that is precisely what we mean by "a quantum harmonic oscillator" as well because we invented the word "harmonic oscillator" in a classical context and when we say "a quantum harmonic oscillator", we precisely mean a quantum theory which looks like a classical harmonic oscillator in (one of) its classical limit(s).

One thing to keep in mind would be that a quantum theory is more basic and thus, it might often be more instructive to think about how a classical theory emerges from a quantum theory than the other way around. Because while we might use a classical theory to second guess a quantum theory out of which the known classical theory emerges, nature certainly works the other way around. Moreover, it might also be useful to notice that there can be quantum theories which admit no, one, or more than one classical limits.

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You are doing both the things: if you would be able to deal with so small energies to distinguish between the different energy levels, then you could take a look at the usual harmonic oscillator in a quantum context. Also, of course when you quantize a system you get an idea of its quantum nature, of which behaviors emerge when you are at extremely low energies.

The motivation to quantize the harmonic oscillator is basically to translate the most simple classical prototype of a physical system to a quantum level. This is particularly useful to build models of atoms, molecules and other quantum systems. So you ultimately do not care too much about the macroscopic object treated in classical mechanics but to its microscopic quantum counterparts. So in this sense maybe the second of your possibilities is the best fitting. But really it is more about model building.

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