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?
Answer
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 ω(n,n−α)=1ℏ{W(n)−W(n−α)}
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)=∞∑α=−∞Xα(n)exp[iω(n)αt]
In modern terminology X(n,n−α) is just the matrix element ⟨n−α|ˆX|n⟩ of the position operator ˆX for the energy eigenstates |n⟩, |n−α⟩
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=−iℏ
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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