= Exponential variational bound for the Yukawa potential
For $V=-Ae^{-\mu r}/r$, the normalized exponential with scale $\alpha$ gives $E(\alpha)=\hbar^2\alpha^2/(2m)-4A\alpha^3/(\mu+2\alpha)^2$. Choosing $\alpha=\mu/2$ proves binding if $\mu<Am/\hbar^2$. Failure to obtain a negative trial expectation outside this range does not prove absence of a <bound state>. At equality a small positive admixture of the scale-$\mu$ exponential has a negative cross matrix element and proves binding too.
Back to article page
