The variational principle gives an upper bound on the ground-state energy without solving the Schrödinger equation exactly. For any normalized trial state $|\psi\rangle$, $$\langle H\rangle_\psi=\langle\psi|H|\psi\rangle\ge E_0,$$ where $E_0$ is the exact ground-state energy.

Choose a family of trial states $|\psi(\alpha)\rangle$ with adjustable parameters. Compute $$E(\alpha)=\frac{\langle\psi(\alpha)|H|\psi(\alpha)\rangle}{\langle\psi(\alpha)|\psi(\alpha)\rangle}$$ and minimize it over $\alpha$. The best value within the chosen family is still an upper bound to $E_0$.

For a one-dimensional Hamiltonian, one might try a Gaussian $$\psi(x;\alpha)=A e^{-\alpha x^2/2}.$$ A narrow Gaussian lowers the potential energy for a confining potential near the origin but raises kinetic energy because rapid spatial variation increases momentum spread. A broad Gaussian does the opposite. Minimization finds the optimal compromise.

The method is powerful because a physically motivated trial state can give useful energies even when the exact eigenfunction is inaccessible. Its limitation is equally important: a poor trial family may produce a loose upper bound while giving no obvious warning about how inaccurate the wavefunction is.

Variational reasoning is central to atomic, molecular and condensed-matter physics, where interacting many-particle Schrödinger equations are rarely exactly solvable.