For a state that has wave function (a pure state), according to the axioms of quantum physics, there is always a complete system of measurement processes that lead, with certainty to certain results. In contrast, for systems that only have a density matrix (the so-called mixed states) there is no complete system of measurements that leads to univocally predictable results.
When an electron interacts with a classical object, a complex system is formed. From the idea of interaction proper it follows that the states of the electron and the apparatus can not be independent. Consequently, the electron that interacts with a measuring device does not have a wave function, but a density matrix. The electron and the measuring device are entangled. How then a pure state evolves?
Not sure if I quite understand what you're driving at - as long as there is an entangled state and decoherence has not set in, then the system (object plus electron) has a wave function.
Be that as it may, the way to obtain in that case meaningful results from measurements would be to use so-called 'protective measurements', which can be brought about in a number of ways. Shan Gao, in his wonderful book 'The Meaning of the Wave Function' (CUP, 2017) describes protective measurements at some length.
I hope this helps.
There are evolution equations for wave functions and evolution equations for density matrices. But that's just words-what matters is the space of states and the operators that act on them.
One can write down evolution equations for either the operators or the states, compute transition probabilities and answer in that way all possible questions it makes sense to ask. The evolution equations for wave function and density matrix are a means to an end-the calculation of transition probabilities-not an end in themselves.
Dear Oscar
I have a very different kind of answer for you. When a particle or particles wavefunction is in a superposition state (and/or a mixture state) it will evolve in complex ways. Keeping them in superposition is the hard part. To collapse to a pure state, just make a measurement. For example, if you measured the total angular momentum of your system, then you'll never get a number that isn't a half-integer number of HBar. It is always exactly one of: 1/2 HBAR, 1 HBAR, 3/2 HBAR, 2 HBAR, ...
To have an exact value for the angular momentum means that you are in a pure state of (total) angular momentum. If you measure the energy with very high accuracy, then afterward the state will be in a pure state along the energy axis. A well defined energy means you're either in one state or the other one, not a superposition any more. That is, a pure state. Make measurements of a complete set of variables, and the state is as pure as can be. That's my answer to your question. But to go on...
So, starting in a complex state (superposition or mixture) many people believe that there is "something" out there is occasionally measuring particle states and causing wavefunctions to spontaneously collapse. No one knows how, or even why, but it happens. Complete collapse leads to a pure state. Incomplete collapse (not all observables are detected) results in something closer to a pure state than the one you started with.
The construction of particles (and the rest of physics) from causal sets is fully discrete. Also, primitive spatial relations are eliminated by exclusive reliance on causal links for order and connection. Thus no continuous waveforms are involved, nor are there any spatial states to which such waveforms could collapse, were they to exist. The constructions were made possible by the discovery that causal sets provide inherent frequency ratios, useful for defining energy ratios and their quantum in accord with E=hf. The inherent metric-- based on ratios-- proves sufficient for physics, where measurement is always by comparison.
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