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

TL;DR

This paper derives the vibrational equations for diatomic molecules using Heisenberg's picture, applying laser stability theory to identify dissociation conditions and minimum oscillation frequencies.

Contribution

It introduces a novel approach by applying laser stability theory to analyze the stability of diatomic molecules in quantum mechanics.

Findings

Diatomic molecules dissociate at unstable equations of motion.

Minimum oscillation frequency at dissociation is determined.

Energy conservation in vibrational motion is demonstrated.

Abstract

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.