Existence of quantum states for Klein-Gordon particles based on exact and approximate scenarios with pseudo-dot spherical confinement

TL;DR

This paper investigates the eigenvalue spectra of Klein-Gordon particles confined in a spherical pseudo-dot potential, analyzing both exact and approximate solutions for constant and variable mass distributions, revealing how mass affects energy levels.

Contribution

It provides a systematic analysis of relativistic eigenvalues for Klein-Gordon particles in pseudo-dot confinement, considering both exact and approximate mass scenarios, which is a novel approach.

Findings

Eigenvalues depend on mass distribution and are larger than 1 fm$^{-1}$ in the exact case.

In the approximate scenario, eigenvalues satisfy E < m_0.

Confluent hypergeometric functions characterize the quantum states.

Abstract

In the present study, Kummer's eigenvalue spectra from a charged spinless particle located at spherical pseudo-dot of the form $r^2+1/r^2$ is reported. Here, it is shown how confluent hypergeometric functions have principal quantum numbers for considered spatial confinement. To study systematically both constant rest-mass, $m_{0}c^2$ and spatial-varying mass of the radial distribution $m_{0}c^2+S(r)$, the Klein-Gordon equation is solved under exact case and approximate scenario for a constant mass and variable usage, respectively. The findings related to the relativistic eigenvalues of the Klein-Gordon particle moving spherical space show the dependence of mass distribution, so it has been obtained that the energy spectra has bigger eigenvalues than $m_{0}=1$ fm$^{-1}$ in exact scenario. Following analysis shows eigenvalues satisfy the range of $E<m_{0}$ through approximate scenario.