Solving the Radial Time-Dependent Schrödinger Equation for a New Class of Coulomb Potentials
Primary Investigator (PI) Name
Eric Stachura
Department
CSM - Mathematics
Abstract
In quantum mechanics, the Schrodinger equation allows us to describe the behavior of a system mathematically, much like how Newton’s laws describe our macroscopic world. However, for a system of many bodies, solving it explicitly quickly becomes a near-impossible task. Time-dependent density functional theory gives us a useful framework for modeling these systems and their properties based solely on the theoretical positions of particles in space. Within this context, examination of the properties of a new class of potentials for a hydrogen atom reduces the time-dependent Schrodinger equation to a system of ordinary differential equations by setting the wavefunction equal to a product of three functions of spherical coordinates r, Θ, Φ. Our focus is on the radial equation, an eigenvalue problem in which exact solutions that satisfy it have yet to be found. Our goal is to correctly determine the change of variables needed to ensure solvability of this equation.
Disciplines
Mathematics | Numerical Analysis and Computation | Ordinary Differential Equations and Applied Dynamics | Quantum Physics
Solving the Radial Time-Dependent Schrödinger Equation for a New Class of Coulomb Potentials
In quantum mechanics, the Schrodinger equation allows us to describe the behavior of a system mathematically, much like how Newton’s laws describe our macroscopic world. However, for a system of many bodies, solving it explicitly quickly becomes a near-impossible task. Time-dependent density functional theory gives us a useful framework for modeling these systems and their properties based solely on the theoretical positions of particles in space. Within this context, examination of the properties of a new class of potentials for a hydrogen atom reduces the time-dependent Schrodinger equation to a system of ordinary differential equations by setting the wavefunction equal to a product of three functions of spherical coordinates r, Θ, Φ. Our focus is on the radial equation, an eigenvalue problem in which exact solutions that satisfy it have yet to be found. Our goal is to correctly determine the change of variables needed to ensure solvability of this equation.