# Jason Barrios

**Major**: Electrical Engineering

**Mentors:** Jacob Levitt, Professor Vlcek

## Interesting Effects of a Particle in a Box

The solution to the Schrödinger equation, the central equation in quantum mechanics (QM), gives us energy values based on specified physical boundary conditions. Simulating these solutions requires specified physical boundary conditions and numerical methods to simplify the large computational cost. The physical boundaries studied here are those of a 1D box potential with finite and infinite potential barriers. The numerical process we use to solve the Schrödinger equation is the finite difference approximation (FDA). We simplify our differential equation into a system of linear equations separated by a finite difference. The FDA creates a grid space where each point on the grid represents part of a solution to the Schrödinger equation. When we increase the number of grid points, the solution converges to the textbook energy eigenvalue solution and takes seconds to run. This allows us an efficient way to visualize the evolution and properties of quantum systems and is a good teaching/learning technique to introduce elementary properties of QM.