Origins of the canonical commutation relation

Question

I have recently been reading Gunter Ludwig's book wave mechanics to get a better understanding of quantum mechanics and in reading through the book I came across the relation $$m\sum_s \{|q_{rs}|^2\omega_{sr}-\omega_{rs}|q_{rs}|^2\}=\hbar$$

Where $\omega_{nm}$ is the quantised frequency $\nu\omega\to\omega_{nm}$, satisfying $\hbar\omega_{nm}=E_n-E_m$ and $q_{nm}$ are the hamiltonian coordinates.

To justify the expression, he points to the equation $J_r=r\hbar+J_0$, where $J$ is the (quantised) action variable, although I am not sure how it follows.

To try and get a better insight, I turned to the paper by Heisenberg, but that just referenced other papers by Khun and Thomas which I could not seem to find online.

Might anyone be able to offer any insight on how this equation was arrived at and explain how experiments were carried out to justify it?

Answer 1

Indeed, Heisenberg's 1925 paper got the ball rolling, but was not recognized as matrix mechanics and transcribed/streamlined in the standard matrix language we all recognize today until later (merely 60 days!) that year in the seminal paper by Born and Jordan Zur Quantenmechanik. Z. Physik 34 858–888 (1925), detailed in Jeremy Bernstein's must-read article Max Born and the quantum theory, American Journal of Physics 73 999 (2005).

Max Born had learned about matrices in his study under Jakob Rosanes at Breslau University. His assistant and former student Pascual Jordan had been an assistant to mathematician Richard Courant at Göttingen, helping in the formulation of Hilbert space! Born got the Nobel prize in 1954 for this work, and for complex and fraught reasons Jordan was skipped over.

Your stated relation, for the quantum oscillator, maps to Born's central relation $\hat x \hat p - \hat p \hat x = i \hbar I$, accessed here; cf details here.

If, on the other hand, you are asking not what it means, in today's terms, but how did Heisenberg intuit it, the word "magic" has been proffered. He is composing radiation intensities in a manner consistent with Bohr's quantization rule, and guessing how $\hat x^2$ and (essentially) $\hat p^2$ behave w.r.t. each other, his "reinterpretation".

A final note added, since, despite all this, it really lies at the heart of the question. The modern understanding and interpretation of the essence of Heisenberg's invention, albeit in the language it created and which the world has adopted since (including the generous term "Heisenberg's equations of motion"), actually got its start in Dirac's 1925 equally breathtaking leap of "reinterpretation" of Heisenberg's paper: The fundamental equations of quantum mechanics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 109 (752), 642-653 (1925), else here.