Wave functions of a particle with polarizability in the Coulomb potential

TL;DR

This paper investigates the quantum behavior of a polarizable scalar particle in a Coulomb potential, deriving a simplified radial equation, analyzing bound states, and calculating the lowest energy level using numerical methods.

Contribution

It introduces a new approach to analyze a polarizable particle in Coulomb fields, deriving a reduced differential equation and demonstrating the existence of bound states with numerical calculations.

Findings

Bound states exist for the system.

The lowest energy level is calculated.

The radial equation involves a -term with Heun functions.

Abstract

Quantum mechanical scalar particle with polarizability is considered in the presence of the Coulomb field. Separation of variables is performed with the use of Wigner $D$-functions, the radial system of 15 equations is reduced to a single second order differential equation, which among the Coulomb term includes an additional interaction term of the form \sigma \alpha^{2} / M^{2}r^{4}. Various physical regimes exist that is demonstrated by examining the behavior of the curves of generalized squared radial momentum P^{2}(r). Eigenstates of the equations can be constructed in terms of double confluent Heun functions. Numerical analysis proves the existence of the bound states in the system; the lowest energy level and corresponding solution are calculated based on generalization of Ritz variational procedure.