Finite-subspace variational method

For an orthonormal trial set ${\varphi }_1,\dots ,{\varphi }_N$, the Rayleigh quotient on their span isIts minimum is the smallest eigenvalue of $\mathcal{H}$, which is therefore the optimal variational upper bound available in that trial subspace.

For the first two sine modes in a width-$a$ infinite well with
$V(x)=9{\hslash }^{2}x/(m{a}^3)$, the Hamiltonian in units ${\hslash }^2/(m{a}^2)$ isIts lower eigenvalue gives the upper bound