2
$\begingroup$

Dirac in his book fundamental of quantum mechanic used the following derivation:

enter image description here

Is this a derivation of the canonical commutation relations (CCR) in quantum mechanics?

$\endgroup$
1

1 Answer 1

3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.