= Gaussian variational bounds for a quartic oscillator
{c}
{title2=$E_0\leq(3\hbar^2/8m)(6mk/\hbar^2)^{1/3}$}
For $H=p^2/(2m)+kx^4$ with $k>0$, the normalized even trial <wavefunction> $f_\alpha=(\alpha/\pi)^{1/4}e^{-\alpha x^2/2}$ has <Rayleigh quotient>
$$
\mathcal E_0(\alpha)=\frac{\hbar^2\alpha}{4m}+\frac{3k}{4\alpha^2}.
$$
The <Gaussian moment recurrence> gives the potential term, and <integration by parts> gives the kinetic term. Its minimum is at $\alpha^3=6mk/\hbar^2$, proving the displayed upper bound by the <Rayleigh-Ritz variational principle>.
An odd trial family $g_\beta=(4\beta^3/\pi)^{1/4}x e^{-\beta x^2/2}$ is orthogonal to the exact even <ground state>. The <Sturm oscillation theorem> identifies the lowest odd state as the first excited state, so minimizing within this family bounds $E_1$:
$$
\mathcal E_1(\beta)=\frac{3\hbar^2\beta}{4m}+\frac{15k}{4\beta^2},\quad
\beta_*^3=\frac{10mk}{\hbar^2},\quad
E_1\leq\frac{9\hbar^2}{8m}\left(\frac{10mk}{\hbar^2}\right)^{1/3}.
$$
Orthogonality to an approximate ground trial alone would not justify this excited-state bound; odd parity ensures orthogonality to the exact ground state. Both bounds exhibit the characteristic quartic-oscillator <energy> scale $(\hbar^4k/m^2)^{1/3}$.
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