Exact Analytical Solution of the N-dimensional Radial Schrodinger Equation with Pseudoharmonic Potential via Laplace Transform Approach

TL;DR

This paper derives exact solutions for the N-dimensional Schrödinger equation with pseudoharmonic and Morse potentials using Laplace transforms, revealing how energy levels vary with dimension and providing a new analytical approach.

Contribution

It introduces a novel Laplace transform method to solve the N-dimensional Schrödinger equation with pseudoharmonic potential and extends the approach to generalized Morse potential.

Findings

Exact bound state solutions obtained for pseudoharmonic potential

Energy eigenvalues vary with the dimension N

Method applicable to generalized Morse potential

Abstract

The second order $N$-dimensional Schr\"odinger equation with pseudoharmonic potential is reduced to a first order differential equation by using the Laplace transform approach and exact bound state solutions are obtained using convolution theorem. Some special cases are verified and variation of energy eigenvalues $E_n$ as a function of dimension $N$ are furnished. To give an extra depth of this letter, present approach is also briefly investigated for generalized Morse potential as an example.