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Canadian Journal of Physics

Corrected non-relativistic bound state solutions in D-dimensional Euclidean space for the Hua potential

A. Khodja, M. Ferdjaoui

Published May 20, 2025• 0 citations
Publisher

Abstract

In this work, we present a rigorous and complete non-relativistic treatment of the problem of a diatomic molecule which moves under the Hua potential in a D-dimensional Euclidean space. The radial Green’s function associated with this potential is constructed within the framework of the Feynman path integral. Different cases of this potential which are distinguished by the values of the deformation parameter q are considered separately. For [Formula: see text] q < 1 and r ∈ ] r0, +∞[, we adopt an appropriate Pekeris-type approximation for the centrifugal potential term to obtain the approximate spectrum of bound states lD ≠ 0. Some numerical results of the energy levels of certain diatomic molecules are calculated by using the correct minimum value of the deformation parameter q given by [Formula: see text]. For q < 0 or [Formula: see text], the problem is solvable only in a three-dimensional space. In both cases, the energy quantization condition is a transcendental equation that can be solved numerically. In the limit [Formula: see text], the energy spectrum and the corresponding wave functions of the radial Morse potential are regained.

DOI
10.1139/cjp-2025-0051

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In this work, we present a rigorous and complete non-relativistic treatment of the problem of a diatomic molecule which moves under the Hua potential in a D-dimensional Euclidean space. The radial Green’s function associated with this potential is constructed within the framework of the Feynman path integral. Different cases of this potential which are distinguished by the values of the deformation parameter q are considered separately. For [Formula: see text] q < 1 and r ∈ ] r0, +∞[, we adopt an appropriate Pekeris-type approximation for the centrifugal potential term to obtain the approximate spectrum of bound states lD ≠ 0. Some numerical results of the energy levels of certain diatomic molecules are calculated by using the correct minimum value of the deformation parameter q given by [Formula: see text]. For q < 0 or [Formula: see text], the problem is solvable only in a three-dimensional space. In both cases, the energy quantization condition is a transcendental equation that can be solved numerically. In the limit [Formula: see text], the energy spectrum and the corresponding wave functions of the radial Morse potential are regained.
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The bound state solution of the Feynman propagator with the deformed generalized Schiöberg potential is determined using an approximation of the centrifugal term. The energy eigenvalue expression is computed using Duru–Kleinert space–time transformation for both positive and negative deformation parameters of diatomic molecules. Besides, the rotation–vibration energy eigenvalues are numerically calculated for some diatomic molecules and compared with those given in the literature. The obtained results are in agreement with those given by state-of-the-art approximate and numerical methods.
In this paper, we present analytic solutions of the Feynman propagator with the Rosen–Morse potential. To this end, an approximation of the centrifugal potential for any l state is used and nonlinear space–time transformations in the radial path integral are applied. A transformation formula that relates the original path integral to the Green function of a new quantum soluble system is derived. Explicit expressions of the bound state energy spectra and the eigenfunctions are obtained and compared to those of Schrödinger formalism. The Eckart potential is also treated as a special case.
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