Exponential variational trial for a square-root half-line well (${\psi }_b(x)=x{e}^{-bx}$)

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