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Izvestiya vysshikh uchebnykh zavedenii Fizika

Exact solutions of the two-dimensional Logunov–Tavkhelidze equation for «delta-circle» potentials in the relativistic configuration representation

Andrei V. Paulenka, Valery N. Kapshai, Yu. A. Grishechkin

Published Jan 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.17223/00213411/68/10/14

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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.
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.
The covariant single-time equations of the quantum field theory are formulated in the relativistic configurational representation. The explicit formulae for the Green functions corresponding to the scattering states are calculated in this representation. Using the derived nonhomogeneous equations the scattering problem is solved exactly for certain potentials (combinations of zero-range potentials). The equations and their solutions are studied in the non-relativistic limit. The conditions of total reflection, available for such potentials, are investigated.
The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.
Abstract Making an ansatz to the wave function, the exact solutions of the D-dimensional radial Schrödinger equation with some molecular potentials, such as pseudoharmonic and modified Kratzer, are obtained. Restrictions on the parameters of the given potential, δ and ν are also given, where η depends on a linear combination of the angular momentum quantum number ℓ and the spatial dimensions D and δ is a parameter in the ansatz to the wave function. On inserting D = 3, we find that the bound state eigensolutions recover their standard analytical forms in literature.
A nonrelativistic equation for the system of two interacting particles within the framework of a model with noncommuting operators of coordinates and momenta of different particles is proposed, and a self-consistent system of equations for the wave function of every quantum state is deduced. Solutions for the lowest states of a hydrogenlike atom are found, and the comparison with analogous solutions of the Klein-Gordon equation for the relativistic spinless problem is performed. In the case where the size of a two-particle system and the Compton wavelengths of particles forming it are of the same order, the essential differences with solutions of the Schrödinger nonrelativistic equation are revealed.