Application of the Nikiforov-Uvarov Method in Quantum Mechanics

Abstract

This book chapter is addressed to readers who want to learn how to solve the time-independent Schrodinger equation (Schrodinger, 1926) in an alternativemethod that was introduced by A. F. Nikiforov and V. B. Uvarov (Nikiforov & Uvarov, 1988). The requirement for understanding the chapter is a knowledge of quantum mechanics in an introductory level and partial differential equations. The primary of the chapter is intended for undergraduate students in physics and chemistry however, it may be used as a reference guide for graduate students and researchers as well. The solution of the Schrodinger equation for a physical system in quantum mechanics is of great importance, because the knowledge of wavefunction Ψ(r, t) and energy E contains all possible information about the physical properties of a system. This knowledge is ranging from the energy, momentum and coordinate of the particle to the wave characteristics of the particle, frequency and wavelength if we describe the quantum mechanical system by the probability amplitude |Ψ(r, t)|2 and its phase (Tang, 2005). Ψ(r, t) is supposed to describe the of a particle subject to the potential energy function V(r), where r represents the spatial position of the particle. For a one-particle, one-dimensional system in cartesian coordinates, we have Ψ(r, t) = Ψ(x, t) and V(r) = V(x) or for a one-particle, three-dimensional system in spherical coordinates, we haveΨ(r, t) = Ψ(r, θ, φ, t) andV(r) = V(r, θ, φ). If wewant to know how the state of the particle changes with time, we need to specify the future state, Ψ(r, t), of a quantum mechanical system from the knowledge of its initial state, Ψ(r, t = 0). To do that an equation postulated by the Austrian physicist Erwin Schrodinger (1887-1961) can help us