= Compactly supported linear trial for hydrogen
{title2=$E(\beta)=5/\beta^2-5/(2\beta),\quad E(4)=-5/16$}
In <atomic units>, use the radial trial <wavefunction> $\psi(r)=\max(1-r/\beta,0)$ with $\beta>0$. Its radial norm integral is $\int r^2\psi^2dr=\beta^3/30$, its gradient contribution is $\int r^2\psi'^2dr=\beta/3$, and its Coulomb contribution is $-2\int r\psi^2dr=-\beta^2/6$. In the hydrogen radial <Rayleigh quotient>, this gives the displayed energy, minimized at $\beta=4$. It is above the exact <hydrogen ground-state energy> $-1/2$, as required by the <quantum variational principle>. The derivative jumps at $r=\beta$, but the function has a square-integrable weak first derivative and is an admissible trial for the energy quadratic form. One must use that form rather than assume that a classical second derivative exists at the join.
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