Is this project an undergraduate, graduate, or faculty project?
Undergraduate
Project Type
individual
Campus
Daytona Beach
Authors' Class Standing
Kassidy Myers, Senior
Lead Presenter's Name
Kassidy Myers
Lead Presenter's College
DB College of Arts and Sciences
Faculty Mentor Name
Keshav Acharya
Abstract
The Schrödinger equation is the foundational equation of non-relativistic quantum mechanics. The hydrogen atom is the simplest system for solving this equation, as it consists of only one proton and one electron. In this project, we work on the Schrödinger equation that models the spherically symmetric states of the hydrogen atom that depend only on the radial coordinate. We simplified and nondimensionalized the radial equation and solved the resulting equation using a power series (Frobenius) method. This approach revealed the physically meaningful solutions and led to quantized energy levels. In addition to finding the analytical solution, we numerically solve the equation for different values of the energy parameter to verify the expected behavior of the system. We then visualize the radial probability distributions, which describe how likely it is to find the electron at different distances from the proton. By plotting these distributions for several energy levels, we can see how the structure of the atom changes with energy.
Did this research project receive funding support (Spark, SURF, Research Abroad, Student Internal Grants, Collaborative, Climbing, or Ignite Grants) from the Office of Undergraduate Research?
No
Included in
Numerical Analysis and Computation Commons, Ordinary Differential Equations and Applied Dynamics Commons, Special Functions Commons
Analytical and Numerical Solutions for the Hydrogen Atom
The Schrödinger equation is the foundational equation of non-relativistic quantum mechanics. The hydrogen atom is the simplest system for solving this equation, as it consists of only one proton and one electron. In this project, we work on the Schrödinger equation that models the spherically symmetric states of the hydrogen atom that depend only on the radial coordinate. We simplified and nondimensionalized the radial equation and solved the resulting equation using a power series (Frobenius) method. This approach revealed the physically meaningful solutions and led to quantized energy levels. In addition to finding the analytical solution, we numerically solve the equation for different values of the energy parameter to verify the expected behavior of the system. We then visualize the radial probability distributions, which describe how likely it is to find the electron at different distances from the proton. By plotting these distributions for several energy levels, we can see how the structure of the atom changes with energy.