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Russian Physics Journal

Solution of the Logunov–Tavkhelidze Equation for the Three-Dimensional Oscillator Potential in the Relativistic Configuration Representation

Yu. A. Grishechkin, V. N. Kapshaǐ

Published Dec 26, 2018• 8 citations
Publisher
DOI
10.1007/s11182-018-1582-4

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We construct approximate analytic solutions of the Logunov–Tavkhelidze equation in the case of a potential that, in the one-dimensional relativistic configuration representation, has the form analogous to the potential of the nonrelativistic harmonic oscillator in the coordinate representation. The wave functions are obtained in both the momentum and relativistic configuration representations. The approximate values of the energy of the relativistic harmonic oscillator are the roots of transcendental equations. The wave functions in the relativistic configuration representation have additional zeros in comparison with the wave functions of the corresponding states of the nonrelativistic harmonic oscillator in the coordinate representation.
We solve exactly the relativistic finite-difference equation for the quantum three-dimensional ring-shaped Quesne oscillator potential. Our investigation is based on a finite-difference version of relativistic quantum mechanics. So-called relativistic configurational r-space is a key concept here. We show that the radial wavefunctions and angular wavefunctions are expressed through the continuous dual Hahn polynomials and Jacobi polynomials, respectively. A discrete energy spectrum has been found. The radial wave functions and energy spectrum have the correct nonrelativistic limit. We also build a dynamical symmetry group SU (1, 1) for the radial part of the equation of motion, which allows us to find the energy spectrum purely algebraically.
In this work, exact solutions are obtained for two-dimensional partial integral equations of the Logunov-Tavkhelidze type for a system of two particles of equal mass in the case of a «delta-circle» potential and a superposition of two such potentials, defined in the relativistic configuration representation. It is shown that for a single «delta-circle» potential, depending on its parameters, the system can have either one bound state or none at all. For a superposition of two «delta-circle» potentials, cases with one or two bound states, or their absence, are possible. Analysis of the wave functions revealed differences in their properties across different representations: in the relativistic configuration representation, the number of zeros of the partial wave function equals the state number of the particle system, whereas in the momentum representation, the number of zeros exceeds that in the configuration representation. The nonrelativistic limit of the obtained solutions is found, which agrees with the solutions of the two-dimensional Schrödinger equation for similar potentials in the momentum and coordinate representations.