Particle in a box

Example

Let $m > 0$ be the mass of a particle, $L > 0$ the length of the box and $\hbar$ the reduced Planck constant. The particle moves on the interval $(0, L)$ between infinitely high walls. The state space is $L^2(0, L)$, and the Hamiltonian is $$H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}$$ with the Dirichlet boundary conditions $\psi(0) = \psi(L) = 0$. Its eigenfunctions, the stationary states, are $$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi x}{L}\right), \qquad E_n = \frac{n^2 \pi^2 \hbar^2}{2 m L^2}, \qquad n = 1, 2, 3, \dots$$

Check: $\psi_n'' = -(n\pi/L)^2 \psi_n$, so $H \psi_n = \frac{\hbar^2}{2m}(n\pi/L)^2 \psi_n = E_n \psi_n$. Both boundary conditions hold, since $\sin 0 = \sin(n\pi) = 0$. The normalisation holds, since $\int_0^L \frac{2}{L}\sin^2(n\pi x/L)\, dx = 1$. These are all eigenfunctions, and they form an orthonormal basis of $L^2(0, L)$, the Fourier sine basis.

So the energy is quantised, every state is a bound state, and the lowest energy $E_1$ is strictly positive. A general solution of the Schrödinger equation is $\psi(x, t) = \sum_n c_n \psi_n(x) e^{-i E_n t/\hbar}$ with $\sum_n \abs{c_n}^2 = 1$. Since $E_n t / \hbar = 2\pi n^2$ at $t = 4 m L^2/(\pi \hbar)$, every state returns to itself at that time.