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Aug 6, 2017 at 20:45 comment added Apollonius @Landau are you asking why these are the correct formulas? If so, then the answer has to do with how you change basis states in a Hilbert space from a position basis to a momentum basis or the other way around.
Aug 6, 2017 at 16:18 comment added user163104 @Landau I am a bit confused as to what you mean by mathematical method that allows. I wrote an answer which I deleted because it was too basic and did not address your full question. You have the intuitive (physical) picture, so which part, (and I mean this without offence, seriously ), of the simple extension of the Fourier series idea to wave packets is the problem? Have you checked with MathSE, as you have the physical idea, and MathSE deals with formal derivations and proofs, if that is what you are after?
Aug 6, 2017 at 15:49 comment added Landau Ok, the intuitive meaning of those formulas was quite clear, but I'm looking for the mathematical method that allows you to write a wave packet in that form.
Aug 6, 2017 at 11:38 history answered Apollonius CC BY-SA 3.0