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I would like to know: did Heisenberg chance upon his Uncertainty Principle by performing Fourier analysis of wavepackets, after assuming that electrons can be treated as wavepackets?

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The route to the uncertainty principle went something like this:

In Heisenberg's brilliant 1925 paper [1], he addresses the problem of line spectra caused by atomic transitions. Starting with the known $$\omega(n, n-\alpha) = \frac{1}{\hbar}\{W(n)-W(n-\alpha) \} $$ where $\omega$ are the angular frequencies, $W$ are the energies and $n, \alpha$ are integer labels, he says basically "let's try to construct a theory which avoids mentioning the electron position $x(t)$ because we can never observe it".

However, in a periodic system (which the electron orbits are) this unobservable quantity for the case where the electron is in the state labelled by n can be Fourier expanded $$ x(n,t)=\sum_{\alpha=-\infty}^{\infty}X_{\alpha}(n)exp[i\omega(n)\alpha t]$$ The Fourier coefficients are labelled by two integers $\alpha$ and $n$, and Heisenberg rewrites these coefficients as $X(n,n-\alpha)$ (Actually he uses the notation $\mathcal{U}$, but $X$ is easier to relate to what it actually is!). This is a quantity with two integer labels, i.e. a matrix. We see that Fourier analysis was intimately involved in Heisenberg's thinking.

In modern terminology $X(n, n-\alpha)$ is just the matrix element $\langle n-\alpha |\hat{X}|n \rangle$ of the position operator $\hat{X}$ for the energy eigenstates $|n \rangle$, $|n-\alpha \rangle$

By applying Heisenberg's matrix representation to the position $X$ and momentum $P$ operators, Born and Jordan [2] were able to derive the commutation relation $$ PX - XP = -i\hbar$$ Heisenberg[3], realizes that this is a splitting up of phase space into cells of dimension $h$ and uses this to deduce an approximate uncertainty principle.

So getting back to the question: no, Heisenberg didn't explicitly arrive at the uncertainty principle by looking at the Fourier analysis of wavepackets, but rather as a consequence of the commutation relations which arose as a consequence of the matrix mechanics he'd discovered. But yes, Fourier analysis was crucial to his reasoning.

Edit: This is a very useful reference for Heisenberg's original thinking on matrix mechanics.

[1]: Heisenberg "Ueber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen" Z. Phys 33 879-893 (1925)

[2]: Born Jordan Zur Quantenmechanik Z. Phys 34 858-888 (1925)

[3]: Heisenbertg Uber den anschaulichten Inhalt der quantentheoretischen Kinematic und Mechanik Z. Phys 43 3-4 172-198 (1927)

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You can listen to Heisenberg remembering the discovery here:

you can say, well, this orbit is really not a complete orbit. Actually, at every moment the electron has only an inaccurate position and an inaccurate velocity, and between these two inaccuracies there is this uncertainty relation.

It was due to a chat with Einstein about the orbits of electrons. There is no mention of a Fourier transform (however it is pretty certain that Heisenberg was quite familiar with its concept).

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