Estimating Ground-State Energy

The previous lesson chose a trial family that happened to contain the exact answer. The variational method earns its keep when the trial family does not contain the true ground state — then it still delivers a rigorous upper bound, and we learn how good (or poor) the guess is.

A deliberately imperfect guess

Take the hydrogen atom, in atomic units ( $\hbar = m_e = e = 1$ , energies in Hartree):

\hat{H} = -\frac{1}{2}\nabla^2 - \frac{1}{r}, \qquad E_0 = -\frac{1}{2} \text{ Ha}.

The true ground state decays exponentially, $\psi_0 \propto e^{-r}$ . Suppose we did not know that and guessed a Gaussian instead,

\psi_\alpha(r) = e^{-\alpha r^2}.

A Gaussian falls off too fast at large $r$ and is too flat at the origin (it has zero slope there, whereas the real wavefunction has a cusp). It is the wrong shape — exactly the situation where the variational principle proves its worth.

The variational energy

Carrying out the (standard Gaussian) integrals for the kinetic and Coulomb terms and normalizing gives

E(\alpha) = \frac{3}{2}\alpha - 2\sqrt{\frac{2\alpha}{\pi}}.

The first term is the kinetic cost of confinement; the second is the (negative) Coulomb attraction. Minimizing,

\frac{dE}{d\alpha} = \frac{3}{2} - \sqrt{\frac{2}{\pi}}\,\frac{1}{\sqrt{\alpha}} = 0 \;\;\Longrightarrow\;\; \alpha^* = \frac{8}{9\pi}.

Substituting back,

E(\alpha^*) = -\frac{4}{3\pi} \approx -0.4244 \text{ Ha}.

Reading the result

Two facts make this a textbook illustration of the method:

The bound holds. $-0.4244 > -0.5$ , so the estimate sits above the exact ground-state energy, exactly as the variational principle demands. A Gaussian can never report an energy below the truth.
The error is modest but visible. The estimate is about $15\%$ too high. A wrong-shape trial state captures the gross balance of kinetic and potential energy but misses the cusp and the slow tail, so it cannot reach the exact value.

Try it

Minimize $E(\alpha) = \tfrac{3}{2}\alpha - 2\sqrt{2\alpha/\pi}$ over $\alpha > 0$ and return the minimum energy (a negative Hartree number). Confirm for yourself that it stays above $-0.5$ .

Run your code to see the quantum state.