An Approximate Solution to the Transcendental Equation Problem of the Finite Square Well Potential in Quantum Mechanics

Abstract

In this paper, we present an approach that gives a formal and an approximate solution for a special class of transcendental equations. This solution is in the form of an infinite series generated by a Taylor reversion process. To showcase this technique, we have chosen the transcendental equation that describes the energy levels of a particle moving in a symmetrical finite square well potential in quantum mechanics. The cases for very deep, very shallow, and the intermediate-sized wells are discussed separately. Our results, when compared with the numerical findings, show the validity of our approach and its potential future application to other similar physics and engineering problems. Keywords: Transcendental equations, Taylor reversion, Algebraic approximation, Finite square well, Bound energy.