In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with respect to the generalised velocity.
My first question is as follows:
Given a quantum mechanical system (specified fully by a quantum mechanical Hamiltonian), and given a generalised "position" operator $Q$ (which may not necessarily be a simple $x$ or $y$ coordinate), is there a systematic process for deriving the quantum-mechanical operator $P$, analogous to the canonical conjugate momentum to $Q$?
My second question is:
Assuming that $P$ exists for a given $Q$, is it true that $[Q, P] = i\hbar$? In the same way that it is true for the special case of the linear momentum operator and the linear position operator?
I keep reading that $[q, p] = i \hbar$ is supposed to be postulate of quantum mechanics. However, if there DID exist a systematic process for deriving P from a given $Q$, then - given a specific $P$, $Q$ (and $H$, if relevant) - you should be able to verify that this is the case.
Thank you in advance for any help you guys can offer. These questions have been completely driving me insane.