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In quantum mechanics, the Hamiltonian Operator is obtained by looking at the classical Hamiltonian and replacing $(x,p)$ with operators $(\hat x, \hat p)$ and for terms involving the product $xp$, the commutation relation is taken into account before replacing.

But classically the Hamiltonian is not unique, how does quantum mechanics take this into account, how can we be sure that we get the same result and equation in QM no matter which Hamiltonian we choose?

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