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Molecular Physics

The stability conditions of diatomic molecules via analogy with the stability theory of lasers

J. Jahanpanah, M. Esmaeilzadeh

Published Mar 8, 2016• 2 citations
Publisher

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 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 energy eigenvalues. The temporal variations of relative position and linear momentum of two oscillating atoms are calculated by solving Heisenberg's equations of motion. The results indicate that the oscillatory behaviour of diatomic molecules are described by the wave packets which consisted of two different frequencies. The stability theory of optical oscillators (lasers) is then exploited to determine the stability conditions of an oscillating diatomic molecule in the different energy levels of Morse potential. It 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.

DOI
10.1080/00268976.2016.1155778

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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 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 energy eigenvalues. The temporal variations of relative position and linear momentum of two oscillating atoms are calculated by solving Heisenberg's equations of motion. The results indicate that the oscillatory behaviour of diatomic molecules are described by the wave packets which consisted of two different frequencies. The stability theory of optical oscillators (lasers) is then exploited to determine the stability conditions of an oscillating diatomic molecule in the different energy levels of Morse potential. It 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.
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.
Excitation of multilevel Rydberg states (atoms and diatomic molecules) is studied under an intense time-periodic perturbation that is instantaneously turned on and off. For a one-quantum excitation from a lower lying state, general expressions for the Laplace images of the population amplitudes of Rydberg states are obtained with regard to their decay characteristics. It is shown that the problem considered is reduced to the determination of the positions and widths of the energy levels of a quantum system in the field of monochromatic laser radiation of the same frequency and intensity as that in a pulse mode. To determine these quantities, an integral formulation is proposed for the eigenvalue problem for energies that is relatively simply solved with regard to the effect of the ionic core and its complex vibration-rotation structure on diatomic molecules. The specific features of the excitation of Rydberg states and the behavior of Rydberg wave packets are studied depending on the intensity and duration of laser radiation. A quantum phenomenon of rotational orientation of electron-excited diatomic molecules is considered.
We show that a direct connection can be drawn, based on fundamental quantum principles, between the Morse potential, extensively used as an empirical description for the atomic interaction in diatomic molecules, and the harmonic potential. This is conceptually achieved here through a non-additive translation operator, whose action leads to a perfect equivalence between the quantum harmonic oscillator in deformed space and the quantum Morse oscillator in regular space. In this way, our theoretical approach provides a distinctive first principle rationale for anharmonicity, therefore revealing a possible quantum origin for several related properties as, for example, the dissociation energy of diatomic molecules and the deformation of cubic metals.
In the recent theory of radioactive decay, and also, as is shown, in the theory of dissociation of a diatomic molecule by the acquisition of rotational energy, there occurs a potential energy curve which has the following shape, as we go, say, from left to right. At the left the potential energy is very high, it then comes down to a minimum, increases to a maximum, and again falls off to an asymptotic value. The problems connected with such a curve are of two types. First, we may be given a particle in the region near the minimum, in an energy level which lies below the maximum, and wish to find the chance that it appear by a quantum mechanical process in the region on the other side of the maximum. Second, we wish to find how the "discrete states" in the neighborhood of the minimum are "broadened" by the continuum on the other side of the maximum. To solve these problems we first find the stationary eigenfunctions. By means of them the width and shape of the "broadened discrete levels" are found immediately. We then use these eigenfunctions to set up a wave packet, or, rather, we show how nature may set up a wave packet, which enables us to solve the first of the problems mentioned. The result justifies the use of complex eigenvalues for the solution of the problem.
We establish for the first time the proper conditions to observe Rabi oscillations between vibrationally dissociative states belonging to different electronic terms of a diatomic molecule. We show by wave packet dynamics simulations that such a process is feasible with the molecular ion Na2+{\mathrm{Na}}_{2}^{+}Na2+​.
The Morse potential quantum system is a realistic model for studying vibrations of atoms in a diatomic molecule. This system is very close to the harmonic oscillator one. We thus propose a construction of squeezed coherent states similar to the one of harmonic oscillator using ladder operators. Properties of these states are analysed with respect to the localization in position, minimal Heisenberg uncertainty relation, the statistical properties and illustrated with examples using the finite number of states in a well-known diatomic molecule.