The quantum harmonic oscillator is an exactly-solvable system where the different representations are easily compared.

For each , this construction is identical to a quantum harmonic oscillator.

The quantum field is an infinite array of quantum oscillators.

A quantum harmonic oscillator has an energy spectrum characterized by:

The simplest such state is the ground state of the quantum harmonic oscillator.

They can also refer specifically to the ladder operators for the quantum harmonic oscillator.

These relations can be used to find the energy eigenstates of the quantum harmonic oscillator.

As in the classical case, the potential for the quantum harmonic oscillator is given by:

Probability density function of a ground state in a quantum harmonic oscillator.

The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem.