There are time dependent & space dependent systems (magnetic fields) and time independent (particle in a box or harmonic oscillator). In the latter the expectation value is the 'average' energy and it works out nicely, but does this apply even if the Hamiltonian is dependent on time? because you don't have a nicely worked out energy value (as with the oscillator) but regardless of what the energy actually looks like is the expectation value the same that is it the total energy?
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2$\begingroup$ Essentially duplicate of: When is the Hamiltonian of a system not equal to its total energy? and Example where Hamiltonian $H\neq T+V=E$, but $E=T+V$ is conserved $\endgroup$– ACuriousMind ♦Apr 24 '16 at 10:05
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$\begingroup$ @ACuriousMind Those questions are about classical mechanics. This one is about QM. The questions are related, but they are not really a duplicate, right? $\endgroup$– AccidentalFourierTransformApr 24 '16 at 10:25
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$\begingroup$ @AccidentalFourierTransform: I don't see how this using quantum mechanical terminology changes anything about the answers - if the classical Hamiltonian is the energy, then so will its quantum version be. If the classical Hamiltonian is not the energy, then neither will its quantum version be. I suspect the question is only phrased in terms of quantum mechanics because the asker has not been exposed to classical Hamiltonian mechanics. $\endgroup$– ACuriousMind ♦Apr 24 '16 at 10:33
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$\begingroup$ @Cleo the classical situation is explained in the book Classical Mechanics by Herbert Goldstein $\endgroup$– jimApr 24 '16 at 18:58
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