= Gaussian trial for a quadratic-plus-quartic oscillator
{c}
{title2=$E_\epsilon(\alpha)=\alpha/2+1/(2\alpha)+3\epsilon/(4\alpha^2)$}
For $H=-d^2/dx^2+x^2+\epsilon x^4$, use the normalized real <wavefunction> $\psi_\alpha=(\alpha/\pi)^{1/4}e^{-\alpha x^2/2}$, $\alpha>0$. The <Gaussian integral> gives $\langle x^2\rangle=1/(2\alpha)$ and $\langle x^4\rangle=3/(4\alpha^2)$, while the kinetic contribution is $\alpha/2$. Thus the <Rayleigh quotient> is the displayed expression. For positive $\epsilon$, its stationary width satisfies $\alpha^3-\alpha-3\epsilon=0$. At $\epsilon=2$ the unique positive solution is $\alpha=2$, giving $E_0\leq13/8$ by the <Rayleigh-Ritz variational principle>. For $\epsilon<0$, taking $\alpha\downarrow0$ makes the negative quartic contribution dominate, so the energy is unbounded below even within this normalized family. The positive-coupling bound and negative-coupling instability concern different signs of the same variational expression.
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