  The vibrational motion equations of both homo and hetero-nuclei diatomic
molecules are here derived for the first time. A diatomic molecule is first
considered as a one dimensional quantum mechanics oscillator. The second and
third-order Hamiltonian operators are then formed by substituting the number
operator for the quantum number in the corresponding vibrational energy
eigenvalues. The expectation values of relative position and linear momentum
operators of two oscillating atoms are calculated by solving Heisenbergs
equations of motion. Subsequently, the expectation values of potential and
kinetics energy operators are evaluated in all different vibrational levels of
Morse potential. On the other hand, the stability theory of optical oscillators
(lasers) is exploited to determine the stability conditions of an oscillating
diatomic molecule.It is peculiarly turned out that the diatomic molecules are
exactly dissociated at the energy level in which their equations of motion
become unstable. We also determine the minimum oscillation frequency (cut-off
frequency) of a diatomic molecule at the dissociation level of Morse potential.
At the end, the energy conservation is illustrated for the vibrational motion
of a diatomic molecule.
