Approximate Energy Spectra of the Quantum Gaussian Well: A Four-parameter Potential Fitting

Abstract

In this work, we present a detailed study of a one-dimensional Schrödinger
equation in the presence of quantum Gaussian well interaction. Further, we investigate the
approximate solutions by using the harmonic oscillator approximation, variational principle, four-parameter potential fitting and numerical solution using the finite-difference method. The parabolic approximation yields an excellent energy value compared with the
numerical solution of the Gaussian system only for the ground state, while for the excited states, it provides a higher approximation. Also, the analytical bound-state energies of the four-parameter potential under the framework of the Nikiforov-Uvarov (NU) method have been used after getting the suitable values of the potential parameters using numerical fitting. The present results of the system states are found to be in high agreement with the well-known numerical results of the Gaussian potential.

Keywords: Gaussian potential, One-dimensional Schrödinger equation, Nikiforov- Uvarov (NU) method, Four-parameter potential.

PACS: 03.65.−w; 02.90.+p; 12.39.Pn.