Time Dependent Schrödinger Equation
Andrew Koller

Description

 Erwin Schrödinger proposed the existence of a differential equation that explains the behavior of quantum sized particles.
This equation led to a paradigm shift in physics and is still used to understand quantum behavior today. Each result and
solution to the equation is a beautiful achievement in physics and students aren't always exposed to the visual representation
of these solutions.

Below is the 3D equation in its simplest form:


where Ψ is the unknown, r is the 3D position, and H is the Hamiltonian operator. Expanding the Hamiltonian operator yields:


where V is the potential energy of the system involved. Essentially, the Schrödinger Equation can be viewed as a statement of
conservation of energy in a quantum system. The solution, Ψ, is a wave easily represented in 2D space. The challenge is to solve
the equation in 3D space and represent the waves in three dimensions.

A 2D particle represented as a wave packet

Goal

 To create a simple visualizer that can visualize a 3D solution given an input of initial conditions of the quantum system. This is
often done with simple initial conditions, say that of a hydrogen atom, but is commonly represented as a static diagram. It is my
goal to create a visualizer that is able to progress through time displaying the behavior of a quantum particle much like the 2D
wave packet above.

2D representations of a hydrogen atom


The above image displays a probability density of where particles will be around the atom. The goal of this project is to be able
to extend this to a simpler system in 3D. The 3D representation will have the ability to vary in time and have a changing energy level.

Timeline