Dirac in his book fundamental of quantum mechanic used the following derivation:
Is this a derivation of the canonical commutation relations (CCR) in quantum mechanics?
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2$\begingroup$ Please do not post images of texts you want to quote, but type it out instead so it is readable for all users and so that it can be indexed by search engines. For formulae, use MathJax instead. $\endgroup$– ACuriousMind ♦Jun 3, 2022 at 0:13
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It is not a derivation per se, more like a suggestive argument. Dirac is assuming that the usual Poisson bracket on the algebra of functions can be replaced by a bracket $\{\cdot,\cdot\}$ on the algebra of operators that satisfies a non-commutative Leibniz rule
$$\{\hat{a}\hat{b},\hat{c}\}~=~\hat{a}\{\hat{b},\hat{c}\}+\{\hat{a},\hat{c}\}\hat{b}.$$
In other words, $\{\cdot,\cdot\}$ is a non-commutative Poisson structure, just like the commutator $[\cdot,\cdot]$. It is perhaps not too surprising that consistency then suggests that $\{\cdot,\cdot\}$ and $[\cdot,\cdot]$ are proportional.
answered Jun 3, 2022 at 0:12