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The quantum harmonic oscillator has played central role of almost every field of physics: quantum field theory, particle physics, quantum mechanics, etc.

I want to know what inspired people through the usage of quantum harmonic oscillator? When did people first realize the power of the quantum harmonic oscillator? I.e., which was the question they were trying to justify? Newton's law of gravity and Coulomb's electronic interaction were written as $\propto\frac{1}{r^2}$, but the interaction strength for the harmonic oscillator is $\propto r^2$.

Surely electronic devices and electrons in molecular orbits might be the inspiration, but how did they generalize such structure, the quantum harmonic oscillator, into so many subjects of physics?

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    $\begingroup$ As for the first time it was used I think it was Planck's black body radiation formula. If I recall correctly he assumed that these oscillators were the ones absorbing and emitting radiation (the whole story can be found in Weinberg's Lectures on Quantum Mechanics, in the introductory chapter). It's within "old quantum theory", but it's QM nonetheless. As for other applications, I'm not entirely sure. $\endgroup$ – S V Oct 30 at 20:29
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    $\begingroup$ @SV Just realized one important notes mentioned in CM two days ago: Bertrand's theorem solved in 1873 (en.wikipedia.org/wiki/Bertrand%27s_theorem ). I guess that was also a strong indication, especially when considering the ideology transition from classical to quantum. $\endgroup$ – ShoutOutAndCalculate Oct 31 at 4:46
  • $\begingroup$ It's very interesting! I did not know about that theorem. Thank you! $\endgroup$ – S V Oct 31 at 17:37
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My 2cents on it is studying harmonic systems was natural because any local or global potential minima can be approximated as a quadratic for modeling purposes. At least for basic quantum mechanics that would be my justification. QFT or particle physics someone else would have to answer the precise why it is powerful. I am aware it is the case, but it is well outside my ballpark.

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