# Chap 39: More About Matter Waves

## Sections

| Sec | Topic |
|-----|------|
| 39-1 | [Energies of a Trapped Electron](39-1-energies-of-a-trapped-electron.ipynb) |
| 39-2 | [Wave Functions of a Trapped Electron](39-2-wave-functions-of-a-trapped-electron.ipynb) |
| 39-3 | [An Electron in a Finite Well](39-3-an-electron-in-a-finite-well.ipynb) |
| 39-4 | [Two- and Three-Dimensional Electron Traps](39-4-two-and-three-dimensional-electron-traps.ipynb) |
| 39-5 | [The Hydrogen Atom](39-5-the-hydrogen-atom.ipynb) |

## Review & Summary

:::{glossary}
Energies of a Trapped Electron
  An electron confined to a one-dimensional infinite potential well of width $L$ has quantized energy levels:

  $$
  E_n = \frac{n^2 h^2}{8mL^2} = \frac{n^2 \pi^2 \hbar^2}{2mL^2}, \quad n = 1, 2, 3, \ldots
  $$ (eq-39-infinite-well)

  where $m$ is the electron mass. The ground state is $n=1$.

Wave Functions of a Trapped Electron
  The spatial wave functions for the infinite well are $\psi_n(x) = \sqrt{2/L} \sin(n\pi x/L)$ for $0 < x < L$. They are normalized and orthogonal. The probability density $|\psi_n|^2$ gives the probability per unit length of finding the electron.

An Electron in a Finite Well
  In a finite potential well, the number of bound states is finite. The wave function penetrates into the classically forbidden region (where $E < U$), with exponentially decreasing amplitude. This penetration affects the energy levels and leads to quantum tunneling.

Two- and Three-Dimensional Electron Traps
  In 2D and 3D traps, the energy depends on quantum numbers for each dimension. For a 3D rectangular box, $E = E_{n_x} + E_{n_y} + E_{n_z}$. **Degeneracy** occurs when different combinations $(n_x, n_y, n_z)$ give the same total energy.

The Hydrogen Atom
  The energy levels of the hydrogen atom are

  $$
  E_n = -\frac{13.6\,\mathrm{eV}}{n^2}, \quad n = 1, 2, 3, \ldots
  $$ (eq-39-hydrogen)

  The ground state is $n=1$. When an electron makes a transition between levels, a photon is emitted or absorbed with energy $\Delta E = hf = |E_i - E_f|$.
:::