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arXiv:1212.3702

Interaction of "rigid" quantum systems

Alexander Kolpakov, Alexander G. Kolpakov

Published Dec 15, 2012• 0 citations
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Abstract

We introduce the notion of a "rigid" quantum system as a system with constant relative positions of its nuclei and constant relative distribution of the electrons with respect to the nuclei. In accordance with this definition, a molecule which does not interact with other objects, is a "rigid" quantum system. Molecule is also "rigid" if it interacts with other objects, but the interaction does not change the intrinsic structure of the molecule (or this change can be neglected). Several "rigid" quantum systems interact one with another in the quantum manner. The interaction is ruled by the Schr{ö}dinger equation [1] written for all the particles of the systems under consideration. We consider the case when the external potential is zero.

DOI
10.48550/arxiv.1212.3702

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We introduce the notion of a "rigid" quantum system as a system with constant relative positions of its nuclei and constant relative distribution of the electrons with respect to the nuclei. In accordance with this definition, a molecule which does not interact with other objects, is a "rigid" quantum system. Molecule is also "rigid" if it interacts with other objects, but the interaction does not change the intrinsic structure of the molecule (or this change can be neglected). Several "rigid" quantum systems interact one with another in the quantum manner. The interaction is ruled by the Schr{ö}dinger equation [1] written for all the particles of the systems under consideration. We consider the case when the external potential is zero.
We use the concept of quantum entanglement to give a physical meaning to the electron correlation energy in systems of interacting electrons. The electron correlation is not directly observable, being defined as the difference between the exact ground state energy of the many-electron Schr ̈odinger equation and the Hartree–Fock energy. Using the configuration interaction method for the hydrogen molecule, we calculate the correlation energy and compare it with the entanglement as a function of the nucleus– nucleus separation. In the same spirit, we analyze a dimer of ethylene, which represents the simplest organic conjugate system, changing the relative orientation and distance of the molecules to obtain the configuration corresponding to maximum entanglement.
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We study certain non-linear generalisations of the Schr{ö}dinger equation which admit static solitonic 2 solutions in absence of external potential acting on the particle. We consider a class of solutions that can be written as a product of a solution of the linear Schr{ö}dinger equation with a peaked quantum soliton, in a regime where the size of the soliton is quite smaller than the typical scale of variation of the linear wave. In the non-relativistic limit, the solitons obey a generalized de Broglie-Bohm (dB-B) guidance equation. In first approximation, this guidance equation reduces to the dB-B guidance equation according to which they move at the so-called de Broglie-Bohm velocity along the hydrodynamical flow lines of the linear Schr{ö}dinger wave. If we consider a spinorial electronic wave function a la Dirac, its barycentre is predicted to move exactly in accordance with the dB-B guidance equation.
Electron scattering by rare-gas dimers is studied for very low incident energies using the zero range potential (ZRP) method. Beyond the traditional ZRP, we consider an alternative formulation accounting for the atomic polarization, inspired in the modified effective range theory. The scattering calculations are reported in fixed nuclei, rigid rotor, and rovibrational approximations, the first two being analytical. An expression for the electron-molecule scattering length is obtained. We find that short-range interactions are the dominant mechanism for rotational transitions in electron scattering by rare-gas dimers, while the long-range interactions may be neglected. Our results show how the elastic, rotational, and rovibrational cross sections depend on the electron-atom scattering length, on the molecular parameters, and also on the inclusion of polarization effects. The principle of detailed balance is discussed in the context of the ZRP method. Finally, we show that the rovibrational coupling considerably affects the rotational cross sections when the rotational constant becomes comparable to the vibrational constant.
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