Vibrational energy level of a diatomic molecule

Vibrational energy level of a diatomic molecule is given by,

$$ E_v = ( v + \frac{1}{2} ) \hbar \omega_0 $$

Where, v=0,1,2,3…. is the vibrational quantum number, \(\omega_0 = \sqrt \frac{k}{\mu} \) , \( \mu \) is the reduced mass of the diatomic molecule and k is the force constant.

The selection rule for transition between vibrational states is,

$$ \Delta v= \pm 1 $$