 Parabolic variational trial for a particle in a box

For an infinite square well of width $a$, the normalized trial $\tilde{\psi }=\sqrt{30/a^5}x(a-x)$ obeys both Dirichlet boundary conditions and has no interior nodes of a wavefunction. Its kinetic energy isThe ratio to the exact ground-state energy is $10/{\pi }^2=1.01321\dots$. Its expansion in the normalized sine eigenfunctions has coefficients $4\sqrt{60}/({\pi }^3{n}^3)$ for odd $n$ and zero for even $n$. This supplies a concrete check of the quantum variational principle and shows that a small energy error can coexist with excited-state admixtures.