Cosine perturbation of the ground level in a one-dimensional box ($E=\frac{{\pi }^2{\hslash }^2}{2m{a}^2}-\frac{ϵ}{2}-\frac{m{a}^2{ϵ}^2}{16{\pi }^2{\hslash }^2}+O({ϵ}^3)$)

For the normalized sine eigenfunctions of a particle in a box on $(0,a)$, multiplying the lowest one by $ϵ\cos (2\pi x/a)$ gives $ϵ({\varphi }_3-{\varphi }_1)/2$. Thus the first-order diagonal shift is $-ϵ/2$, and the only second-order coupling is to level three. With $E_3=9E_1$, its contribution is $({ϵ}^2/4)/(E_1-E_3)=-{ϵ}^2/(32E_1)$, giving the displayed expansion. This simple selection rule makes the full second-order perturbation theory sum a single term.