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][1] 
 - [Coupled Q-oscillators as a model for vibrations of polyatomic molecules, D.Bonatsos, C.Daskaloyannis, The Journal of Chemical Physics 106, 605 (1997)][2]
 - [Quantum groups and their applications in nuclear physics, D.Bonatsos, C.Daskaloyannis, Progress in Particle and Nuclear Physics, v.43, 1999, p. 537-618][3] (see also [here][4] 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 ][5]
 - [Dynamical algebra of the q‐deformed three‐dimensional oscillator, J. Van der Jeugt, J. of Math. Phys. 34, 1799 (1993)][6]
 - [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][7]

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. 


  [1]: http://iopscience.iop.org/article/10.1088/0305-4470/24/15/001
  [2]: http://aip.scitation.org/doi/citedby/10.1063/1.473189
  [3]: http://www.sciencedirect.com/science/article/pii/S0146641099001003?via%3Dihub
  [4]: https://arxiv.org/pdf/nucl-th/9909003.pdf
  [5]: http://iopscience.iop.org/article/10.1088/0305-4470/24/20/009/meta
  [6]: http://aip.scitation.org/doi/abs/10.1063/1.530138
  [7]: http://iopscience.iop.org/article/10.1088/0305-4470/24/15/002/meta