Sections#

Review & Summary#

Energies of a Trapped Electron#

An electron confined to a one-dimensional infinite potential well of width \(L\) has quantized energy levels:

(382)#\[ E_n = \frac{n^2 h^2}{8mL^2} = \frac{n^2 \pi^2 \hbar^2}{2mL^2}, \quad n = 1, 2, 3, \ldots \]

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

(383)#\[ E_n = -\frac{13.6\,\mathrm{eV}}{n^2}, \quad n = 1, 2, 3, \ldots \]

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