= Exponential variational trial for a square-root half-line well
{title2=$\psi_b(x)=xe^{-bx}$}
The trial $xe^{-bx}$, $b>0$, satisfies the wall condition in the <half-line square-root potential well>. Its norm is $1/(4b^3)$, kinetic <energy> is $\hbar^2b^2/(2m)$ and <potential energy> is $(15a/16)\sqrt{\pi/2}\,b^{-1/2}$. The optimal $b$ satisfies $b^{5/2}=(ma/\hbar^2)(15/32)\sqrt{\pi/2}$. At the minimum, $\langle V\rangle=4\langle T\rangle$ and $E_{\rm var}=5\hbar^2b^2/(2m)=1.876493E_*$. The <quantum variational principle> makes this an upper bound on the <ground-state energy>.
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