Exact variational solution in a two-level system

If a normalized trial family covers the entire two-dimensional Hilbert space, minimizing its Rayleigh quotient gives the exact lower eigenvalue rather than merely an upper bound.

The eigenvalues of $\left(\begin{array}{cc}{E}_1& h\\ h& E_2\end{array}\right)$ are their mean plus or minus $\sqrt{(E_2-E_1{)}^2/4+h^2}$, so nonzero coupling increases their separation.