# Wavefunction normalization

How do we normalize a wavefunction that's a linear combination of sines and cosines (or of $Ae^{ikx}+Be^{-ikx}$ -- they're the same, right)? One you square it, wouldn't the integrand be oscillating through all space, and thus infinite?

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 Could you please specify which "things" are supposed to cancel? The norm surely can't cancel because it is a sum of positively semidefinite terms. Your function is a combination of plane waves so much like a single wave, its normal can't be one. One could normalize similar combinations to a wave function if the $A/B$ ratio were determined as a function of $k$ and if it were continuous. In that case, $|A|^2+|B|^2$ would be normalized exactly in the same way as $|A|^2$ is normalized if there's only one term, one plane wave. – Luboš Motl May 13 '11 at 4:45 You wouldn't normalize it over all space, the wavefunction would be zero, or something else, outside of a finite range. – Ramashalanka May 13 '11 at 4:46 @Luboš I'm sorry, I was forgetting to square it. But now that I remember that, it seems psi^2 is infinite? – wrongusername May 13 '11 at 4:48 @Ram I thought that for scattering states in a finite square well, psi to the left of the well is A e^ikx + B e^-ikx, which would be some infinite area to integrate over. – wrongusername May 13 '11 at 4:50 @wrongusername: scattering states aren't normalized, instead we look for reflection and transmission ratios. Bound states in a finite square well will decay outside the well like $e^{-k x}$ (with real $k$). – Ramashalanka May 13 '11 at 4:54
$$\langle\psi|\psi\rangle = \int_{-\infty}^{\infty} |A e^{ikx} + B e^{-ikx}|^2\mathrm{d}x = \infty$$
(EDIT: just thought I should mention that $\ldots = \infty$ doesn't mean the integral literally equals infinity, it's a notation for "does not converge".)