Quantum vibration

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

The Hamiltonian of the particle is:

where m is the particle's mass, k is the force constant, ${\displaystyle \omega ={\sqrt {\frac {k}{m}}}}$ is the angular frequency of the oscillator, ∧x is the position operator (given by x), and ∧p is the momentum operator, (given by ${\displaystyle {\hat {p}}=-i\hbar {\partial \over \partial x}\,.}$) The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy.

...
Wikipedia