Math Bits - Particle in a 2D Box

Prerequisites

Students should be able to:

Write down the energy eigenfunctions and eigenvalues for a 1D infinite square well.
Use Fourier series to decompose an arbitrary function into sines (and cosines).
Construct the full time-dependent wave function given an initial wave function.

Solving the Schrodinger Equation in 2 Dimensions (Lecture, 25 minutes)
Degeneracy (SGA - 25 min)
- Each group starts with a different relation between $a$ and $b$, then find:
  1. The energy of the ground state and the corresponding state/ket.
  2. The full wave function (including time dependence) of the ground state.
  3. The energy of at least five excited states and the corresponding state/ket.
    - And their degeneracy if there is more than one state that goes with each energy.
Two-dimensional Box (SGA - 40 min)
- For $a=b$, each group gets a different state/ket:
  1. Write the full wave function (including time dependence) of your state.
  2. Plot $\Psi(x,y,0)$ and $\vert\Psi(x,y,0)\vert^2$. Animate them as time progresses.
- After everyone reports out, if time, everyone should repeat for a superposition.
Sturm-Liouville Theory (Lec - 15 min)

Placeholder question: Particle on a finite cylinder.