Hydrogen atom

Example

Model the hydrogen atom as one electron in the Coulomb field of a fixed proton. Use atomic units, in which the electron mass, the elementary charge, $\hbar$ and $4\pi\varepsilon_0$ all equal $1$. The Hamiltonian is the Schrödinger operator $H = -\frac12 \Delta - \frac{1}{\abs{x}}$ on $L^2(\R^3)$, where $\Delta$ is the Laplacian. Its essential spectrum is $[0, \infty)$. Below $0$ its spectrum consists of the eigenvalues $$E_n = -\frac{1}{2n^2}, \qquad n = 1, 2, 3, \ldots,$$ and $E_n$ has multiplicity $n^2$.

Check for $n = 1$: let $\psi(x) = \pi^{-1/2} e^{-r}$ with $r = \abs{x}$. For a radial function, $\Delta f = f'' + \frac{2}{r} f'$, so $\Delta \psi = \left( 1 - \frac{2}{r} \right) \psi$. Then $H\psi = -\frac12 \psi + \frac1r \psi - \frac1r \psi = -\frac12 \psi$. Also $\int_{\R^3} e^{-2r}\, dx = 4\pi \int_0^\infty r^2 e^{-2r}\, dr = \pi$, so $\psi$ has norm $1$.