In previous post I mentioned that the First Commutation Relation and Second Commutation Relation are equivalent. Here is a proof from Classical Perspective.
We assume one of them to be true. For example, we assume First Commutation Relation to be true. i.e.
Classical Hamiltonian is given by
We next compute the commutation between Hamiltonian and Position
We see the operator in the last term is precisely the velocity operator
Therefore from definition of velocity
So we conclude that
It then follows
(Note it’s only proved for non-relativistic hamiltonian, i still haven’t worked out the most general case at the moment)