Variational method

Remarks

By the spectral theorem, $\ip{\psi}{H\psi} = \int_{\sigma(H)} \lambda \, d\mu_\psi(\lambda)$, where $\mu_\psi$ is the spectral measure of $\psi$, of total mass $\norm{\psi}^2$. Since $\lambda \ge E_0$ on $\sigma(H)$, the integral is at least $E_0 \norm{\psi}^2$, with equality exactly when $\mu_\psi$ is concentrated at $E_0$. Vectors whose spectral measure is concentrated near $E_0$ make the quotient as close to $E_0$ as one likes.

Example: for the hydrogen atom $H = -\frac12 \Delta - \frac{1}{\abs{x}}$ on $L^2(\R^3)$, with $E_0 = -\frac12$, try the Gaussians $\psi_\alpha(x) = e^{-\alpha \abs{x}^2}$, $\alpha > 0$. A computation gives the quotient $\frac{3\alpha}{2} - 2\sqrt{\frac{2\alpha}{\pi}}$, which is smallest at $\alpha = \frac{8}{9\pi}$, with value $-\frac{4}{3\pi} \approx -0.424$. This is an upper bound for $E_0 = -0.5$, as the theorem says, and it is not sharp because the true ground state $e^{-\abs{x}}$ is not a Gaussian.

The min-max theorem extends the principle to the higher eigenvalues below the essential spectrum. Restricting $\psi$ to a finite-dimensional subspace gives the Rayleigh–Ritz method.

Going deeper

For a student

The method turns the eigenvalue problem into a minimisation problem. In practice one chooses a family of trial functions with a few parameters and minimises the energy over the parameters. The Hartree–Fock method is the variational method with Slater determinants as trial functions, and the min-max theorem gives the excited levels in the same spirit. The method only gives upper bounds; lower bounds need other tools, such as Temple's inequality.