There has been quite a lot of literature on the applications of $q$-numbers, $q$-derivatives, $q$-deformations, etc, of various algebraic models of physics. Such applications range from deformations of simple harmonic oscillator(s) to the development of quantum groups and their applications in field theories.
Just to mention a few papers (my ex-phd supervisor has quite some work on the field):
- Generalized deformed oscillator and nonlinear algebras, C Daskaloyannis 1991 J. Phys. A: Math. Gen. 24 L789
- Coupled Q-oscillators as a model for vibrations of polyatomic molecules, D.Bonatsos, C.Daskaloyannis, The Journal of Chemical Physics 106, 605 (1997)
- Quantum groups and their applications in nuclear physics, D.Bonatsos, C.Daskaloyannis, Progress in Particle and Nuclear Physics, v.43, 1999, p. 537-618 (see also here for the arxiv version).
- The many-body problem for q-oscillators, E G Floratos, Journal of Physics A: Mathematical and General, Volume 24, Number 20, 1991
- Dynamical algebra of the q‐deformed three‐dimensional oscillator, J. Van der Jeugt, J. of Math. Phys. 34, 1799 (1993)
- WKB equivalent potentials for the q-deformed harmonic oscillator, D Bonatsos, C Daskaloyannis and K Kokkotas, J. of Phys. A: Math. and Gen., Volume 24, Number 15, 1991
A significant amount of similar literature can be found at the Journal of Mathematical Physics, J. of Physics A: Mathematical and general, Communications of Mathematical Physics, etc.