Gaussian variational estimate for a solvable sextic potential
= Gaussian variational estimate for a solvable sextic potential
For the trial state $e^{-\alpha x^2/2}$, the dimensionless Rayleigh quotient is
$$
\varepsilon(\alpha)=2+\frac\alpha2-\frac3{2\alpha}+\frac{15}{8\alpha^3}.
$$
Its unique minimizer is
$$
\alpha_*^2=\frac{-3+\sqrt{54}}2,
$$
and at the minimum $\varepsilon(\alpha_*)=2+2\alpha_*/3-1/\alpha_*$.