In the harmonic approximation the potential energy is a quadratic function of the normal coordinates. Solving the Schrödinger wave equation, the energy states for each normal coordinate are given by
where n is a quantum number that can take values of 0, 1, 2 ... In molecular spectroscopy where several types of molecular energy are studied and several quantum numbers are used, this vibrational quantum number is often designated as v.
The difference in energy when n (or v) changes by 1 is therefore equal to , the product of the Planck constant and the vibration frequency derived using classical mechanics. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency  (in the harmonic oscillator approximation).
See quantum harmonic oscillator for graphs of the first 5 wave functions, which allow certain selection rules to be formulated. For example, for a harmonic oscillator transitions are allowed only when the quantum number n changes by one,
but this does not apply to an anharmonic oscillator; the observation of overtones is only possible because vibrations are anharmonic. Another consequence of anharmonicity is that transitions such as between states n=2 and n=1 have slightly less energy than transitions between the ground state and first excited state. Such a transition gives rise to a hot band. To describe vibrational levels of an anharmonic oscillator, Dunham expansion is used.