Vibrational quantum number

Quantum mechanics

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 $${\displaystyle E_{n}=h\left(n+{1 \over 2}\right)\nu =h\left(n+{1 \over 2}\right){1 \over {2\pi }}{\sqrt {k \over m}},}$$ 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.