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I am trying to have better understanding of localized wave functions. Apparently free particle de Broglie waves are NOT normalizable and act as delocalized functions which was the original rationale behind their use for explanation of electrons in double slit experiment. so far so good !

But then we use Fourier transform to make them localized that they can represent a particle and behave normalizable . Here is what I can not understand: if wave function gets localized ( say to the size of an electron - which actually is supposed to be point-like ) , then such wave can not cover the distance between both slits to be able to make interference patterns in the double slit experiment, .... and if wave function remains delocalized , then it will not be able to get normalized and consequently its wave function will not present accurate probability amplitude .

The only way I can think of this is to have a localized wave function that is substantially larger than the slits distance but vanishes at the infinity, but it will be much much larger than a particle - let alone a theoretically point-like particle.

My question is: when making a localized wave function, how big the particle size or wave area should be to make interference patterns and be normalizable at the same time? are there known limits for that Fourier section?

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  • $\begingroup$ The wave function for an electron begins even before the electron is emitted from the anode ...i.e. the excited electron/atom is already interacting with the EM field of the apparatus. These EM forces can be described as virtual forces. With the Feynman path integral you can compute a probability function .... the electron will localize somewhere in that pattern at the screen. Similar to photons dark areas have no "localizations", the bright areas get all the photons. $\endgroup$ Nov 12, 2023 at 15:32

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A wave function is not a particle. It is a probability amplitude that represents where you are likely to detect the particle and allows us to compute the probability.

Therefore, a wave function is not point-like. If one would force it to be point-like at one instance of time, then when time evolves it would immediate starts to disperse.

The notion that an electron is a point is based on the fact that an electron interacts with other fields in a point. So it is actually the interaction that is point-like.

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  • $\begingroup$ would you please expand your views on this question as well ? $\endgroup$
    – hyportnex
    Nov 12, 2023 at 12:21

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