Quantum Mechanics – Chapter 13: Variational Principle

13.1 The Variational Method

Goal: an approximate (upper-bound) estimate of the ground-state energy.

Procedure

Choose a normalized trial function \(\psi(x,\alpha)\) with a variational parameter \(\alpha\).
Compute \(E(\alpha)=\left\langle T\right\rangle+\left\langle V\right\rangle\): \[E(\alpha)=\int\psi^*\Big(-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}\Big)dx+\int\psi^*V\psi\,dx.\]
Minimize: \(\dfrac{dE}{d\alpha}=0\) at \(\alpha_0\), with \(\dfrac{d^2E}{d\alpha^2}\Big|_{\alpha_0}>0\).
\(E_0\approx E(\alpha_0)\).

Variational theorem

For any normalized \(\psi\): \(E_{\rm exact}\le\left\langle \psi\middle|H\middle|\psi\right\rangle\). Equality holds only if \(\psi\) is the exact ground state.
Enlarging the trial family (more parameters, containing the old family) can only lower (or keep) the estimate.