Singular inverse square potential in coordinate space with a minimal length

TL;DR

This paper investigates how a minimal length, introduced via a generalized uncertainty principle, regularizes the singular inverse square potential in quantum mechanics, providing analytical solutions and insights into bound states.

Contribution

It presents an analytical solution to the deformed Schrödinger equation with a minimal length for the inverse square potential, showing regularization effects and deriving the energy spectrum.

Findings

Minimal length regularizes the potential's singularity.

Analytical solutions expressed in confluent Heun functions.

Energy spectrum consistent with existing literature.

Abstract

The problem of a particle of mass m in the field of the inverse square potential is studied in quantum mechanics with a generalized uncertainty principle, characterized by the existence of a minimal length. Using the coordinate representation, for a specific form of the generalized uncertainty relation, we solve the deformed Schr\"odinger equation analytically in terms of confluent Heun functions. We explicitly show the regularizing effect of the minimal length on the singularity of the potential. We discuss the problem of bound states in detail and we derive an expression for the energy spectrum in a natural way from the square integrability condition; the results are in complete agreement with the literature.