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However, normally in a textbook, quantization of a system only involves commutation relations between generalized coordinates and generalized momentums. for example, [itex][q, p] = ih[/itex]

With that in mind, I have the following questions.

1. Does other poisson bracket relations(not only between generalized coordinates and generalized momentums) hold in quantum mechanics? If so, why none of them is mentioned in a textbook

2. Normally, textbooks consider other mechanical quantities as functions of generalized coordinate and generalized momentum. In this way, the classical definitions are transformed into operator relations. Generally, definitions are equations. So the question comes up naturally that whether all classical equations can be regarded as quantum operator equations.

This is definitely not true in Schrodinger picture 'cause some of the equations contain derivatives of mechanical quantities and operators are time-independent in Schrodinger picture. What if in Hisenberg picture?