Exact Quantization Rule to the Kratzer-Type Potentials: An Application to the Diatomic Molecules

TL;DR

This paper derives exact analytical solutions for the Schrödinger equation with Kratzer-type potentials in D dimensions, providing explicit energy levels and wave functions for diatomic molecules, and validates results with numerical comparisons.

Contribution

It introduces a simple exact quantization rule method to solve the hyperradial Schrödinger equation for Kratzer potentials, yielding precise energy levels and wave functions for diatomic molecules.

Findings

Exact energy levels calculated for various diatomic molecules.

Normalized wave functions derived analytically.

Results agree with other established methods.

Abstract

For any arbitrary values of $n$ and $l$ quantum numbers, we present a simple exact analytical solution of the $D$-dimensional ($D\geq 2$) hyperradial Schr% \"{o}dinger equation with the Kratzer and the modified Kratzer potentials within the framework of the exact quantization rule (EQR) method. The exact energy levels $(E_{nl})$ of all the bound-states are easily calculated from this EQR method. The corresponding normalized hyperradial wave functions $% (\psi_{nl}(r))$ are also calculated. The exact energy eigenvalues for these Kratzer-type potentials are calculated numerically for the typical diatomic molecules $LiH,$ $CH,$ $HCl,$ $CO,$ $NO,$ $O_{2},$ $N_{2}$ and $I_{2}$ for various values of $n$ and $l$ quantum numbers. Numerical tests using the energy calculations for the interdimensional degeneracy ($D=2-4$) for $I_{2}, $ $LiH,$ $HCl,$ $O_{2},$ $NO$ and $CO$ are also given. Our results obtained by EQR are in exact agreement with those obtained by other methods.