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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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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