I have been trying to understand why the term quantum 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 Why are quantum groups so called? this one Intuition behind the definition of quantum groups, and this answer of Pavel Etingof on mathoverflow What is the relation between quantum symmetry and quantum groups?.

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.