The quantum linear harmonic oscillator, and hence coherent states, arise in the quantum theory of a wide range of physical systems.
2.
A very important use of complex numbers is for solving the differential equation of a linear harmonic oscillator : d ^ 2y / dx ^ 2 + y = 0.
3.
The energy eigenstates of the linear harmonic oscillator ( e . g ., masses on springs, lattice vibrations in a solid, vibrational motions of nuclei in molecules, or oscillations in the electromagnetic field ) are fixed-number quantum states.