Vibration Energy Levels of an Harmonic Oscillator

When a molecule is undergoing simple harmonic oscillations, the restoring force of the harmonic vibrations is directly proportional to the displacement and the energy of the harmonic oscillator is given by
                                      
Where is the vibration quantum number, which may be zero or an integer and is the equilibrium frequency of vibration of the oscillator i.e. for small displacements.

Strictly speaking the movement of  a real oscillator is not perfectly harmonic. This is because as the displacement increases, the restoring force becomes weaker and for large enough amplitude of vibration, the atoms of the molecule must fall apart i.e. the molecule must dissociate into atoms. Such a real oscillator is said to be an enharmonic oscillator. P.M. Morse in oscillator by including one additional term as follows:
                                
Where Xe is called the anharmonicity constant.

For more help in Vibration Energy Levels of an Harmonic Oscillator click the button below to submit your homework assignment