#### Project Title

### Solving the Radial Time-Dependent Schrödinger Equation for a New Class of Coulomb Potentials

#### Disciplines

Mathematics | Numerical Analysis and Computation | Ordinary Differential Equations and Applied Dynamics | Quantum Physics

#### Abstract (300 words maximum)

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.

#### Academic department under which the project should be listed

CSM - Mathematics

#### Primary Investigator (PI) Name

Eric Stachura

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.