Treatment of the quasi-harmonic potential with the centrifugal type term in the Schroedinger equation via Laplace transform

TL;DR

This paper presents the first analytic solution to the Schrödinger equation with a quasi-harmonic potential including a centrifugal term, using Laplace transform methods, providing explicit energy eigenvalues and wave functions.

Contribution

The authors introduce a novel application of Laplace transform to solve the Schrödinger equation with a complex potential including a centrifugal term, deriving explicit solutions.

Findings

Analytic expressions for energy eigenvalues and wave functions obtained.

Approximation of the critical orbital quantum value and its upper bound.

First-time solution of the quasi-harmonic potential problem in quantum mechanics.

Abstract

In the quantum frame, for 3-dimensional space, in the two body problem case, we approach the Schr\"odinger equation (SE) taking in account the potential: Vq(r)=Dr^2+(A/r)+(B/r^2) called by us quasi-harmonic potential with the centrifugal type term, with D,A,B >0 and D<<1. We use Laplace transform method (LTM) and we find for the first time an analytic solution of the Vq-potential problem. Namely, using directly and inverse Laplace transformations, we obtain the complete forms of the energy eigenvalues and wave functions. Furthermore, for this potential Vq, we make considerations about critical orbital quantum value "lc" and we obtain a useful approximation of upper bound "lc+" to "lc".