Hi guys, I've been studying the problem of the simple, one-dimensional quantum mechanical pendulum of length [itex] \ell [/itex] and mass [itex] m [/itex]. We first apply the small-angle approximation which of course reduces the problem to the simple harmonic oscillator. This part is easy enough. However, we then are asked to use perturbation theory to solve for the first order energy correction due to the invalidity of the small-angle approximation. My confusion is this: a true eigenfunction on a ring must have periodic boundary conditions imposed. However, the small-angle approximation in this problem yields eigenfunctions which certainly aren't periodic on the circle. I guess this is okay, after all, we're looking only at small angles. However, perturbation theory requires integrals over the *entire* configuration space of the coordinate. So in my case, do I integrate from 0 to 2pi, even though the un-perturbed eigenfunctions aren't really eigenfunctions over this space? I'm tempted to integrate over the real line instead, but this also doesn't sound quite okay.