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

Stability conditions of diatomic molecules in Heisenbergs picture: inspired from the stability theory of lasers