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My question is basically this..

In regular math, Fourier Coefficients give the "amount" a particular frequency is available in any periodic signal. The squares of sum of coefficients is not equal to 1 in regular Math course.

But in Quantum Mechanics I'm told that any function can be represented by sum of functions of infinite square well and the sum of squares of magnitudes of coefficients is equal to one(obviously the probability of finding the particle is equal to 1).

Why this discrepancy? What am I missing? All I know is basically every periodic functions is sum of sinusoidal signals with Fourier coefficients...

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It is when the function is assumed to be normalized (which is often in QT), i.e.

$$ \int_a^b |\psi|^2 dx = 1. $$

This implies similar constraint on the Fourier series coefficients.

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    $\begingroup$ the mathematical reason behind this is contained in Parseval's theorem or more generally Plancherel's theorem en.wikipedia.org/wiki/Plancherel_theorem $\endgroup$
    – Wihtedeka
    May 12 at 7:57
  • $\begingroup$ You mean this is true only in QM? If the infinite square well solutions are used to represent another quantum function, then their sum of coefficients is 1... but if we use these solutions to represent a real world function outside of the QM, then this need not be the case as the normalization is not enforced in such functions...right? $\endgroup$
    – Siddaram
    May 12 at 23:52
  • $\begingroup$ Yes, if there is no requirement for normalization, then the sum of squared magnitudes of coefficients need not be 1. $\endgroup$
    – Ján Lalinský
    May 13 at 13:52

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