Exact solution to the Schrodinger's equation with pseudo-Gaussian potential

TL;DR

This paper derives exact solutions for the Schrödinger equation with a pseudo-Gaussian potential, calculating bound and metastable state energies and wave-functions, and analyzing transmission through the potential barrier.

Contribution

It provides the first exact analytical solutions for the Schrödinger equation with pseudo-Gaussian potential, including bound and metastable states.

Findings

Exact eigenfunctions and energy levels for bound states

Analysis of metastable states and transmission coefficients

Explicit wave-functions for the pseudo-Gaussian potential

Abstract

We consider the radial Schr\" odinger equation with the pseudo-Gaussian potential. By making an ansatz to the solution of the eigenvalue equation for the associate Hamiltonian, we arrive at the general exact eigenfunction. The values of energy levels for the bound states are calculated along with their corresponding normalized wave-functions. The case of positive energy levels, known as meta-stable states, is also discussed and the magnitude of transmission coefficient through the potential barrier is evaluated.