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Toward chemical applications of Heaviside Operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise 1

Valentino A. Simpao

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

This brief note presents the exact analytical solution of an example class of dynamical systems, realized by differential equations. These problems occur in quantum mechanics and their generalizations in theoretical physics, but they particu- larly find application to chemical systems. Addressed is the exact analytical solution of the radial Schrodinger equation for the nonrelativistic N-particle system with pair- wise 1

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•Dec 9, 2024•0 citations
This brief note presents the exact analytical solution of an example class of dynamical systems, realized by differential equations. These problems occur in quantum mechanics and their generalizations in theoretical physics, but they particu- larly find application to chemical systems. Addressed is the exact analytical solution of the radial Schrodinger equation for the nonrelativistic N-particle system with pair- wise 1
Toward chemical applications of Heaviside Operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise 1rij{\frac{1}{r_{ij}}}rij​1​ radial potential in quantum phase space
Toward chemical applications of Heaviside Operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise 1rij{\frac{1}{r_{ij}}}rij​1​ radial potential in quantum phase space
It is well known that the exact solutions play an important role in quantum mechanics since they contain all the necessary information regarding the quantum model under study. However, the exact analytic solutions of nonrelativistic and relativistic wave equations are only possible for certain potentials of physical interest. In this paper, bound state solutions of the Schrodinger and Klein-Gordon equations with Modified Hylleraas plus attractive radial potentials (MHARP), have been obtained using the parametric Nikiforov-Uvarov (NU) method which is based on the solutions of general second-order linear differential equations with special functions. The bound state eigen energy solutions for both wave equations were obtained. Also special cases of the potential have been considered and their energy eigen values obtained.
The Schrödinger equation is a linear partial differential equation that describes the temporal evolution of the state of a quantum mechanical system. One of the pillars of quantum mechanics, the Schrödinger equation has been used to describe a wide variety of phenomena such as wave-particle duality, atomic orbitals, quantum oscillations, and even some fluid mechanics problems. Despite its established importance in the development of quantum mechanics, a solution of the Schrödinger equation for an arbitrary distribution of quantum mechanical potential or for an arbitrary number of space dimensions remains unknown. This work seeks to rectify this theoretical roadblock by providing an analytical solution through a recasting of the well-known Cole-Hopf transformation which is used to linearize a broad class of PDEs of which the Schrödinger equation becomes a member after an earlier change of variables. These two successive transformations render the general wave function proportional to the solution of a forced heat equation, allowing an explicit analytical solution applicable to any system in any number of space dimensions. Two practical applications to deriving the explicit wave functions of multielectronic atoms and to the forced Burgers equation of fluid mechanics are discussed, enabling greater mathematical insight into quantum entanglement and nonlinear fluid systems.