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 nuclear physics and  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