# Origins of the canonical commutation relation

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?

• This is probably best asked on Physics.SE rather than history as it's way too specialised. Feb 1, 2021 at 8:09

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".