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If we find an expression for the total energy of a system in terms of classical mechanics, can we replace the observables with their quantum-mechanical operators and state that this new equation acting on the wave function should give you the energy eigenvalues? My gut reaction is to say no, because there must be some quantum mechanical effects which just simply can't be accounted for in classical mechanics, however I noticed when working on solving a rigid rotor that it is indeed the case. Moreover, isn't the Hamiltonian in QM derived by following this prescription?

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# Classical Energy vs Quantum Energy

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