Demystifier said:
A recent related paper:
http://lanl.arxiv.org/abs/1609.05041
It points out that the standard conservation law is only a statistical law, which, by itself, is not sufficient to understand conservation of energy at the individual level.
Thanks for sharing this! In the case where the particle is measured with a high final energy, I think this shows the same effect we were discussing: the
total energy after the measurement is higher than the initial energy, with the difference made up by a loss of energy in "branches" of the WF where the particle was not measured- if you believe in those.
The article itself focuses on the change in the probability distribution for the particle's energy, under unitary evolution. I would point out one detail that they didn't mention: If we look for eigenstates of the full Hamiltonian (particle + opener +interaction) , rather than of the particle and opener separately, it is clear that some of these will have superpositions of the free and trapped particle. In those eigenstates that make up the initial conditions in the article, and that are such superpositions, the free element in fact has high energy, which, I think, implies that the trapped element does as well. Meanwhile other, purely trapped eigenstates have lower energies. So this gives us an alternative decomposition of the initial WF of the trapped particle, one which does have high-energy terms, in contrast to the simple Fourier decomposition which does not.
In other words, you cannot know what the possible energies are for the particle in the initial state without knowing the details of interactions it may have in the future, because even if it's currently not entangled with anything, there are many ways to write its WF as a sum of frequencies, and the "correct" one depends on all possible interaction Hamiltonians. Is this correct?