The Variational Method

Most quantum systems cannot be solved exactly, but there is a remarkably simple way to bound the ground-state energy from above: guess a wavefunction. The variational theorem guarantees that for any normalized trial state, the expectation $\langle H\rangle$ is greater than or equal to the true ground-state energy $E_0$, with equality only if your guess happens to be the exact ground state. So you build a trial wavefunction with an adjustable knob, compute the energy, and slide the knob to push $\langle H\rangle$ as low as it will go; the minimum is your best estimate, and you know it is an honest upper bound. This playground does it for the hydrogen atom using a Gaussian trial $e^{-a r^2}$, the same shape that underlies the Gaussian basis sets of every quantum-chemistry code. The catch is visible in the scene: the true ground state $e^{-r}$ has a sharp cusp at the nucleus, a kink that a smooth Gaussian simply cannot reproduce. Because of that mismatch the best Gaussian lands at $\langle H\rangle = -0.424$ Hartree, a respectable but clearly imperfect approximation to the exact $-0.5$. The lower panel plots $\langle H\rangle$ against the trial width $a$: the curve dips to a minimum at $a^\ast = 8/9\pi$ and, no matter how you tune the parameter, never crosses the forbidden line at $E_0$. That floor is the whole content of the variational principle.