In the fall semester of 2022, we worked with Ethan Decker on the perturbation of decoherence-free subspaces. A decoherence-free quantum dynamics refers to a quantum system that, despite the interactions with its environment, still evolves as a unitary dynamics. Ethan conducted perturbation analysis and studied how a small perturbation can lead to a departure
from the subspace.

Project title: The Calculation of Transition Pathways

Team members: Lilith, Mohmmad and Xikai

Background of the problem: Many problems in chemistry involve meta-stable states, where the system stays for certain period of time. An important problem is the transition mechanism from one meta-stable state to the other. The path of such transition is known as the minimum energy path, and the mathematical model is a differential equation with boundary conditions.

Mathematical methods: We use B-spline functions as basis functions, and solve the differential equations using a nonlinear least square algorithm.

Summary of results: We have implemented the algorithm in MATLAB, and studied a cluster of atoms modelled by the Lennard-Jones potential.  Poster Presentation by the group

My name is Xiantao Li, a math professor at Penn State. In the past few years, I have offered a few projects to undergraduate students to have fun with. So check it out! There might be something that interests you.