Rayleigh-Ritz variational principle

The extremal Rayleigh quotients of a self-adjoint operator are the extremal spectral values. In particular, a Hamiltonian with a discrete spectrum satisfies $R[\psi ]\ge E_0$, with equality precisely on its ground-state eigenspace. More generally, minimizing over states orthogonal to the first $j$ eigenspaces gives an upper bound on $E_j$.

For a one-dimensional even potential, the ground state is even and the first excited state is odd. Every normalized odd trial wavefunction is therefore orthogonal to the ground state and gives

Forand the trial state ${\psi }_a(x)=e^{-a{x}^2/2}$,This is negative for some $a>0$ for every $V_0>0$, proving that the attractive well has a bound state. For small $V_0$, taking $a=V_0^2$ gives

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