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A simple question. In classical mechanics, the Hamiltonian is not always equal to the total energy of the system, cf. e.g. this Phys.SE post. Is this also the case in quantum mechanics, or does something 'fix' it in this case?

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No, discrepancies between the Hamiltonian and the total energy in a quantum system is more or less taken over from the corresponding classical system.

Examples:

  • The Hamiltonian in non-relativistic mechanics doesn't know about the relativistic rest energy $mc^2$ of the particles.

  • The Hamiltonian relative to some local reference frame doesn't know about the total energy relative to some fiducial inertial frame, cf. e.g. section 8.2 in H. Goldstein, Classical Mechanics.

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  • $\begingroup$ I'm not sure if thats the sort of 'not accounting for total energy' that I meant. I meant cases analogous to the famous forced oscillator problem in classical mechanics $\endgroup$
    – Alex Gower
    Dec 17, 2021 at 13:59
  • $\begingroup$ Do you have some reference in mind? Which page? $\endgroup$
    – Qmechanic
    Dec 17, 2021 at 14:15
  • $\begingroup$ Hmm maybe I just mean for driven systems? $\endgroup$
    – Alex Gower
    Dec 17, 2021 at 14:24

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