# Integer physics

Are there interesting (aspects of) problems in modern physics that can be expressed solely in terms of integer numbers? Bonus points for quantum mechanics.

Yes, although it depends on how one interprets the terms "aspects" and "solely". Take, for example, the Hamiltonian for the one-dimensional quantum harmonic oscillator. The spectrum of the Hamiltonian (set of possible energies for the oscillator) is $$E_n = (n+\tfrac{1}{2})\hbar\omega, \qquad n = 0,1,2,\dots$$ The spectrum is discrete, and the eigenvalues can be put in one-to-one correspondence with the set of non-negative integers (aka the set of eigenvalues is countable). In fact, there is a large class of Hamiltonians in quantum mechanics with discrete spectra, and for each such Hamiltonian the same comments about the spectrum and integers applies.
I would say that a nice example of this is the Quantum Hall Effect. Here the conductivity $\sigma$ come in integer multiples of $e^2/h$. In this case, the integers here have a topological origin.