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?