Formula Method for Bound State Problems

TL;DR

This paper introduces a straightforward formula to solve bound state problems in quantum mechanics, applicable to various wave equations, providing explicit energy spectra and wave functions with demonstrated accuracy and efficiency.

Contribution

The paper develops a universal formula for bound state solutions of quantum wave equations, simplifying calculations and extending applicability to multiple potential models.

Findings

Accurate energy spectra and wave functions derived using the formula.

Simplified method compared to traditional approaches.

Validated against existing eigenvalue problems.

Abstract

We present a simple formula for finding bound state solution of any quantum wave equation which can be simplified to the form of $\Psi"(s)+\frac{(k_1-k_2s)}{s(1-k_3s)}\Psi'(s)+\frac{(As^2+Bs+C)}{s^2(1-k_3s)^2}\Psi(s)=0$. The two cases where $k_3=0$ and $k_3\neq 0$ are studied. We derive an expression for the energy spectrum and the wave function in terms of generalized hypergeometric functions $_2F_1(\alpha, \beta; \gamma; k_3s)$. In order to show the accuracy of this proposed formula, we resort to obtaining bound state solutions for some existing eigenvalue problems in a rather more simplified way. This method has been shown to be accurate, efficient, reliable and very easy to use particularly when applied to vast number of quantum potential models.