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I need to make an exercise related to quantum mechanics. (Specifically I need to apply Fermi's golden rule where the initial and final states are both plane waves).

The system is 1 dimensional, infinite and the potential is zero everywhere. This means that $\psi = e^{\pm ikx}$ are solutions to the Schrodinger equation with energies $E = \frac{\hbar^2 k^2}{2m}$.

My question is, how do I normalize these plane waves when the space is infinite?

(Normally I would just say the normalization constant is equal to $\frac{1}{\sqrt{L}}$, but L is infinite here).

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    $\begingroup$ Plane waves cannot be normalised. However, wave packets can be. $\endgroup$ – lemon Jun 14 '16 at 11:47
  • $\begingroup$ Look up delta-function normalization. Plane waves are not normalizable as functions on the real axis, but are always normalizable as functionals or distributions. Note: not statistical distributions, but generalized functions . $\endgroup$ – udrv Jun 15 '16 at 1:52

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