Electrons are often thought of as little particles, but a more precise statement of the nature of electrons is that they move like waves and exchange energy and momentum like particles. When an electron is confined by a potential $V(x)$, it is described by a wavefunction. The wavefunction is a solution to the Schrödinger equation,

$$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} +V(x)\psi(x) = E\psi(x).$$

Here $m$ is the mass of the electron, $E$ is the energy of the electron, and $\psi$ is the wavefunction. The probability of finding the electron at position $x$ is $\psi^*\psi$.

One of the few analytically solvable problems in quantum mechanics is the infinite potential well, also known as a particle in a box. For this problem, the potential $V(x)$ is defined to be zero in the interval $0 < x < L$ and infinite otherwise.

$$ V(x) = \begin{cases} 0 & 0 < x < L\\ \infty & \textrm{otherwise} \end{cases}$$

In one dimension, the solutions are

$$\psi_n (x) = \begin{cases} \sqrt{\frac{2}{L}} \sin \left( \frac{n \pi x}{L} \right) & 0 < x < L \qquad n=1,2,3,\cdots\\ 0 & \text{otherwise} \end{cases} $$

Solutions can only be found at discrete energy levels. The energies can be found by substituting the solutions into the Schrödinger equation,

$$ E_n = \frac{\hbar^2 \pi^2}{2mL^2} n^2.$$

Some of the wave functions are plotted below. \(n=\) 1 2 3 4 5 , \(m=\) kg, \(L=\) m, 

\(E\) [eV]		\( \psi\)
	\(x\) [Å]

As $n$ increases, the spacing between the energy levels increases because of the $n^2$ factor in the expression of the energy.