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arXiv:2204.00445

Alternative solution to a quantum-mechanical four-particle system in one dimension

Francisco Marcelo Fernández

Published Mar 31, 2022• 0 citations
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Abstract

We propose an alternative solution to a quantum-mechanical four-particle system in one dimension with two- and three-particle interactions. The solution of the eigenvalue equation in center-of-mass and Jacobi coordinates is considerably simpler than a procedure proposed recently based on spherical coordinates.
DOI
10.48550/arxiv.2204.00445

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In this article, a four-particle quantum system is introduced with energy discrete spectrum including a harmonic potential and a three-body interaction potential. By defining the Jacobi coordinates for each particle, separately, one coordinate is eliminated as a transition in the energy spectrum. Then the system is studied in polar coordinates and by using the variables separation method, the schrödinger equation of the system is transformed into three separate differential equations. Therefore, energy eigenvalues and wave eigenfunctions are calculated in each dimension. Also, The wave eigenfunctions figures are investigated in one and three dimensions.
The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.