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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
up vote 8 down vote accepted

You don't. Sinusoidal wavefunctions like these, a.k.a. plane waves, are non-normalizable, because the integral which defines the norm does not converge.

$$\langle\psi|\psi\rangle = \int_{-\infty}^{\infty} |A e^{ikx} + B e^{-ikx}|^2\mathrm{d}x = \infty$$

(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".)

People tend to talk about these plane waves because it's easy to figure out how they behave, but in reality, a wavefunction is never just a pure plane wave. It's always some linear combination of plane waves with different frequencies/wavelengths, such that the overall wavefunction goes to zero at large distances quickly enough to make the integral converge. For example, you might have a plane wave confined within a potential well, so that the wavefunction is zero everywhere outside the well, or you could have a Gaussian wavepacket.

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So when Marder (Condensed Matter Physics, 2nd) writes that solutions of the free-particle Schrödinger equation are $$\Psi_{\vec{k}} = \frac{1}{\sqrt{V}}\mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{r}}$$ he implies that we'll be using LCs of them? Or is it an error? $$\left[\frac{\vec{r}}{V}\right]_{-\infty}^\infty$$ is either indefinite or infinity. He states that the plane wave solutions are normalized, which bothers me... maybe I'm missing something in his argument. – CHM Sep 29 '13 at 18:32
You always use a normalizable linear combination of plane waves, i.e. a wavepacket, to represent a real particle. (NB technically an infinite sum is not a linear combination, but it's close enough for physics purposes.) If $V=0$ everywhere, then a plane wave is not normalizable in the sense I describe in the answer, but in a bounded space it can be normalized. Maybe that's what the book is talking about. Alternatively, Marder could be using a different norm. (Or it could be an error. I can't tell without seeing the context.) – David Z Sep 29 '13 at 19:28

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