= Exponential trial state for a half-harmonic oscillator
{title2=$\psi_\alpha=xe^{-\alpha x},\quad\Lambda=\alpha^2+3/\alpha^2\geq2\sqrt3$}
For the dimensionless <half-harmonic oscillator> $-\psi''+x^2\psi=\lambda\psi$ on $x>0$ with $\psi(0)=0$, the trial family $xe^{-\alpha x}$, $\alpha>0$, is admissible for the <Rayleigh-Ritz variational principle>. The moments $\int_0^\infty x^ne^{-2\alpha x}dx=n!/(2\alpha)^{n+1}$ give norm squared $1/(4\alpha^3)$, derivative energy $1/(4\alpha)$, and potential energy $3/(4\alpha^5)$. Its <Rayleigh quotient> is $\alpha^2+3/\alpha^2$, minimized at $\alpha=3^{1/4}$. The resulting upper bound $2\sqrt3$ exceeds the exact lowest eigenvalue three; the exact ground-state shape is $xe^{-x^2/2}$.
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