q-difference equations and quantum mechanics 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?
Pavel Etingof says that "the main mechanisms through which quantum groups appear in physics is the same as for usual Lie groups" but I have seen hundreds of physics books with applications of classical Lie groups (classical groups usually, $SO,U,SU,...$) to quantum mechanical problems but none for quantum groups.
Also when did the term q-calculus, quantum calculus, q-hypergeometric series first appear -in particular in what order did they appear?
Thank you.
 A: Regarding the first part of the question: 

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?

In my understanding, the answer is yes, there have been lots of applications in various different settings coming from quantum mechanical problems. As far as i know, the first applications of this kind were the algebraic descriptions of the solutions of the Schrodinger equations for (deformed) $q$-oscillators and $q$-rotators together with the computations of the corresponding energy spectrums, transition rates etc. Among the pioneering papers: 


*

*The quantum group $SU_q(2)$ and a q-analogue of the boson operators, L C Biedenharn, 
J. of Phys. A: Math. and Gen., vol 22, 18, L873, 1989

*On q-analogues of the quantum harmonic oscillator and the quantum group $SU(2)_q$, A J Macfarlane, J. of Phys. A: Math. and Gen., vol 22, 21, 4581, 1989


Since then, there has been quite a lot of literature on similar topics. If we confine ourselves to low and intermediate energy QM (that is: mainly models of atomic and nuclear physics), an early overview can be found at: 


*

*Quantum groups and their applications in nuclear physics, D.Bonatsos and C.Daskaloyannis, Prog. in Part. and Nucl. Phys., Vol 43, 1999, Pages 537-618


For a more complete list and further discussion on "actual uses of $q$-calculus and quantum groups ... ", maybe you will be interested in the answers (and the references included) to the following question: 
Is there any published physics article where $q$-mathematics is applied? 
A: There exist applications of q-calculus to physics, but there is no direct relation to quantum mechanics. You can find an overview of some of these applications in q-Calculus and physics (paywall).
This should not be a surprise, because the "q" in q-calculus does not stand for "quantum", at least it did not originally. It predates quantum mechanics by two centuries and it is believed that the "q" was originally used (by Euler?) as an abbreviation of "quotient". The history is described by T. Ernst in The different languages of q-calculus.
See also this discussion at HSM

Sources: The cited article by Ernst says (on page 39): "Euler had in fact already introduced q-series and Jacobi continued to use the letter q, which has survived until today."
As far as I could check, Euler used $x$ instead of $q$ (for example, in Evolutio producti infiniti), but Jacobi did indeed use $q$. Here is Jacobi's q-series from  Fundamenta nova theoriae functionum ellipticarum (1829, page 86):

The systematic theory of q-series started with E. Heine (1847).
