LittleSchwinger said:
TL;DR Summary: Why is the fundamental evolution of systems postulated to be unitary rather than a more general CPTP map?
The usual justification for why the evolution of physical systems is unitary in quantum mechanics involves arguments like "probabilities must sum to 1" or similar arguments that apply equally to any CPTP map. I'm just curious what justifications people here would use for selecting out unitary evolution in particular.
To be clear this is really for gathering ideas on teaching. I'm not disputing unitary evolution or anything like that and am aware of some justifications for postulating fundamental unitary evolution. By necessity I am assuming responses are familiar with CPTP maps.
Such subtleties are usually answered by the "symmetry approach". I must admit, the more I think about, how to teach introductory quantum mechanics the more desparate I get, because you cannot use these arguments, which involves the theory of Lie groups and Lie algebras as well as their ray representations on a Hilbert space, which are pretty advanced mathematical concepts. Of course you cannot in any way derive quantum mechanics from classical mechanics in a deductive way, because quantum mechanics is by far the theory with the larger realm of applicability, and classical mechanics is an approximate effective description for much coarse-grained macroscopic observables of many-body systems, but that's also a can of worms of course.
For me the most plausible heuristics is, assuming you can argue with group theory, that you look for a quantum theory thats dynamics is compatible with the symmetries of Newtonian spacetime, and it needs to be only the part of the corresponding Galilei group that is smoothly connected with the identity. From the general framework of QT this implies that you look for unitary ray representations of the Galilei group, and that's most easily done by first looking for such representations for the corresponding Lie algebra.
Then it turns out that the unitary ray representations that lead to a satisfacoritly interpretable quantum dynamics can be lifted to a unitary representation of an extended Lie algebra, which substitutes the rotation subgroup by SU(2) (its covering group) and has the one non-trivial central charge ##\neq 0##, and this central charge represents the mass of the system.
The by definition the time evolution is realized by the time translations with the Hamiltonian as the generator, and since we have lifted the ray representations without loss of generality to unitary representations of the (somewhat extended quantum) Galilei group, the time evolution is a unitary transformation. You can also show that you can define a position observable in the proper sense.
You find a very good treatment of the representation theory of the Galilei algebra (group) in Ballentine, Quantum Mechanics.
Of course, this approach cannot be used as an introductory framework for the QM 1 lecture. Here I have no better idea to start with a historical approach, emphasizing the failure of Bohr-Sommerfeld quantization, and then use the "wave-particle dualism" of "old quantum mechanics" (emphasizing from the beginning that this is fact is a dead end and one of the reasons why modern QM has been worked out pretty soon!) to motivate the Schrödinger equation as a wave equation describing de Broglies "matter waves", which of course from the very beginning have to be interpreted in the probabilistic way a la Born. From this you can motivate the Hilbert-space formulation with the representation of the observables by self-adjoint operators.
Then you can also use the Diracian approach of "canonical quantization", assuming you students have heard about Hamiltonian analytical mechanics and Poisson brackets in the mechanics lecture. Then for the classical point particle you simply start with ##\{\cdot,\cdot \} \rightarrow [\cdot,\cdot]/\mathrm{i} \hbar## and the "Heisenberg algebra" for position and momentum (for the beginning motion in 1 dimension is enough) with ##[\hat{x},\hat{p}]=\mathrm{i} \hbar \hat{1}##. From this you can construct position and momentum eigenstates by using the fact that ##\hat{p}## is the generator for spatial translations, i.e., it has the same mathematical meaning as in classical mechanics, where ##p## also is the generating function for an infinitesimal canonical transformation, describing spatial translations.
