Apologies in advance for what is not a short read. I hope someone will find it interesting enough to answer. Alternatively, if my misunderstandings are infuriating enough to warrant a response outlining my misconceptions, that would also be a positive outcome for me.
I was reading ‘Energy Non-Conservation in Quantum Mechanics’ (see here for the abstract) which, in addition to being a plug for the many-worlds interpretation of QM, also discusses some subtleties related to energy conservation, and allegedly shows that energy is NOT conserved, even for the system taken as a whole (with some caveats around what exactly is meant by 'the system').
The basic setup the authors describe seems uncontroversial - it is clear that in a very specific sense measurements in QM are non-unitary operations, and that after measuring a general state that was originally in an energy superposition, the energy you measure will not, in general be the same as the expectation value of the energy before the measurement. This doesn’t seem like a big deal to me - we can say the system doesn’t have a well-defined energy until you measure it, in the same sense that we say that of particle’s and their momenta/position/etc.
They go on to say:
In most discussions of quantum measurement [lack of energy conservation] is obscured, either because the system being measured is treated as an open system interacting with the outside world, or because the interaction Hamiltonian is modelled as time-dependent (in either of which cases, nobody should expect energy to be conserved). In terms of the current example, we might imagine that there is some principle whereby differences in the energies of the environment states ${ \left| i_e \right> }$ precisely compensate for energy differences in the system states.
The piece with my added emphasis is what I have always assumed is the case. The crux of their paper appears to be the claim that the construction they introduce shows this is not the case:
We can check that there is no such principle, and that environmental effects do not generally restore energy conservation, by constructing [on page 3 to 4 of the paper linked above] a simple but explicit model of a closed system with a complete time-independent Hamiltonian.
This begs the question: Under what conditions, if at all, can a quantum mechanical model of a closed system with a time-independent Hamiltonian be mapped to a model of an open system, and/or a system with a time-dependent Hamiltonian? I feel like the answer should be ‘quite often, if not always’, but this is a vague intuition that I don’t know how to capture in the language of ordinary QM (and is also possibly completely wrong).
Related (but different) question: "Conservation of energy, or lack thereof," in quantum mechanics
