2.1.2 Quantum Oscillators

Application of quantum mechanics to the diatomic system leads to several important conclusions:

1. The system cannot be observed at any random vibrational energy between $-E_{max}$and $E_{max}$as depicted in Section 2.1.1.
2. The vibrational energy of a given state is related to the classical frequency of oscillation.
   
   $$E_{v}=\left(v+\frac{1}{2}\right)h\nu$$
   
   
   where v is the vibrational quantum number, h is Planck's constant and $\nu$ is the classical frequency of oscillation. In other words, the possible energies for a system are discrete, as given by states described by the vibrational quantum number v.
3. The energy between two discrete states is also discrete. The energy required to change states corresponds to the discrete energy between states.
4. The vibrational state can be changed by absorption (or emission) of infrared light, provided:
   • The light frequency is equal to the frequency of the equivalent classical oscillator.
   • The transition between the states is allowed (See Section2.1.4).