I learnt in some Wikipedia articles that the terms of the P and Q matrices designed by Heisenberg were composed of Fourier coefficients. Could you provide some explanation on how these coefficients were related to frequency radiations measurements of the hydrogen atom? Did Heisenberg have access to physical experimentation to corroborate his theory's development?
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$\begingroup$ Please link to the Wikipedia articles in question. $\endgroup$– SpencerNov 28, 2021 at 20:37
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$\begingroup$ In general, on all Stack Exchange sites, rather than "I read somewhere that [the claim in your own words]", it's far better to say "I read in [link to web page]: [direct quotation taken from that citation]. [Optional: your rewording of your interpretation of the text].". That allows others to verify the original details, and to confirm that your understanding of it is correct. (In most cases it is correct, but without the citation, no one can be sure.) $\endgroup$– Ray ButterworthNov 30, 2021 at 14:35
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$\begingroup$ I was refering to the following Wikipedia article en.wikipedia.org/wiki/Matrix_mechanics where I read: The frequencies that make up the outgoing wave are then integer multiples of the orbital frequency, and this is a reflection of the fact that X(t) is periodic, so that its Fourier representation has frequencies 2πn/T only. $\endgroup$– JCRCanNov 30, 2021 at 23:14
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The basic "old quantum theory" facts, such as the Bohr model, fitting the phenomenological Rydberg formula $E_n \propto 1/n^2$ were common knowledge among research physicists at the time, despite deep-seated conceptual malaise. WP details:
According to the Maxwell theory the frequency $\nu$ of classical radiation is equal to the rotation frequency $\nu_{rot}$ of the electron in its orbit, with harmonics at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levels $E_n$ and $E_{n−k}$ when $k$ is much smaller than $n$. These jumps reproduce the frequency of the k-th harmonic of orbit $n$. For sufficiently large values of $n$ (so-called Rydberg states), the two orbits involved in the emission process have nearly the same rotation frequency, so that the classical orbital frequency is not ambiguous.
In formulas, you get an almost linear formula in k, so, then, harmonics of a single frequency, (just like the linear harmonic oscillator featured later in Heisenberg's paper!), $$ \Delta E\propto {-1\over n^2} - {-1\over (n-k)^2}= \frac{2kn+k^2}{n^2(n-k)^2} \approx {2k\over n^3}. $$
Heisenberg reinterpreted, essentially out of his magic hat, the quantities involved in these upper-level transitions of the hydrogen spectrum, aggressively avoiding invocation of unobservables such as electron trajectories, which were a disquieting feature of the old quantum theory.
He modeled the system as a "black box" with arbitrary formal features obeying a strange noncommutative sudoku, that produce these linear spacings in the spectrum. The WP article you are citing identifies these infinite matrices with the routine oscillator matrices of today.
