📦 Particle in a box

The Infinite Square Well (ISW) is an idealization of a physical system that strictly prohibits the particle from straying beyond a certain range of x-values, but exerts no influence over the particle within those limits. So the particle is “free” to roam, but only over a prescribed range on the x-axis. — Visualizing Quantum Mechanics with Python

One-dimensional case

🧠 3D visualization based on the book Visualizing Quantum Mechanics with Python
🧠 2D visualization based on SquareWell.html by Daniel V. Schroeder
🐍 A VPython demo is available as well, see infinite_squarewell.py

📌 Highly recommended background information: Simulating quantum mechanics with Python
👉 Note that the energy of the eigenstates go like $n^2$.
👉 For example, we observe that the $n = 2$ state has four times the energy, and therefore four times the frequency of the ground state $n = 1$!
👉 More physics software by Daniel V. Schroeder can be found here

Two-dimensional case

A 2D Infinite Square Well (ISW) is a potential well that has zero potential energy over a finite domain in two directions, say the $x$- and $y$-directions, and is infinite outside that domain. The simplest case is a rectangular domain in the $xy$-plane with sides $L_x$ and $L_y$. In this case the 2D time-independent Schrödinger wave equation factors into two 1D time-independent Schrödinger wave equations, one in the $x$-direction and one in the $y$-direction. The wavefunction is then a product of the two 1D wavefunctions. The wavefunction is then given by a superposition of energy eigenstate wavefunctions of the form $\psi_{nm}(x, y) = A \sin(n\pi x/L_x) \sin(n\pi y/L_y )$. — Visualizing Quantum Mechanics with Python

What am I looking at?

🧠 Idea taken from the book Visualizing Quantum Mechanics with Python
👉 Highly recommended background information: Visualizing complex numbers and wavefunctions in one and two dimensions

📌 Analytic time evolution for a particle in a two-dimensional infinite square well.
📌 The well is represented by the stationary basis $\psi(n,m) = sin(n\pi x/L_x) sin(m \pi y/L_y)$.
📌 The wavefunction evolves by applying the corresponding phase factors.