I have been trying to understand why the term [quantum][1] is so easily accepted for calculus based on q-numbers $[n]_q=\frac{q^n-1}{q-1}$ and q-analogs of classical operators (derivatives, integrals,...).

But the best hints I could find is this question
https://mathoverflow.net/questions/227889/why-are-quantum-groups-so-called
this one
https://mathoverflow.net/questions/204051/intuition-behind-the-definition-of-quantum-groups?noredirect=1&lq=1,
and this answer of Pavel Etingof on mathoverflow
https://mathoverflow.net/questions/16024/what-is-the-relation-between-quantum-symmetry-and-quantum-groups/16158#16158.

But I could not find any attempt to connect the q-calculations arising from the mathematical idea of deformation (in algebra or combinatorics) to precise concepts in quantum mechanics. Have there ever been actual uses of q-calculus and quantum groups to computing or understanding solutions of Schrodinger equations, or functions actually arising from (physical) quantum mechanics, like states of the harmonic oscillator or some simple atomic hamiltonian like that of the hydrogen atom?

Also when did the term q-calculus, quantum calculus, q-hypergeometric series first appear -in particular in what order did they appear?

Thank you.


  [1]: https://en.wikipedia.org/wiki/Quantum_calculus