Lowest-order correction for the bendulum

[eman2009]

what is the lowest-order correction to the ground state for the pendulum with small angle?

 

[per.sundqvist]

$$cos(x)=1-x^2/2!+x^4/4!-...$$
$$\delta E\propto<\Psi_0\mid x^4\mid\Psi_0>$$