First-order ground state of the quartic oscillator
= First-order ground state of the quartic oscillator
For a <quantum harmonic oscillator> perturbed by $\lambda X^4$, write $X=\sqrt{\hbar/(2m\omega)}(A+A^\dagger)$. Since
$$
(A+A^\dagger)^4|0\rangle=3|0\rangle+6\sqrt2|2\rangle+2\sqrt6|4\rangle,
$$
<first-order nondegenerate perturbation theory> gives
$$
\boxed{|0_\lambda\rangle
=|0\rangle-\frac{\hbar\lambda}{4m^2\omega^3}
\left(3\sqrt2|2\rangle+\sqrt{\frac32}|4\rangle\right)+O(\lambda^2).}
$$