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No, it was my request to construct a POVM for this joint position-momentum measurement in the hope, how the abstract, mathematical concept for predicting probabilities from the quantum formalism in a more general way than the straight-forward standard one, is to be constructed for a real physical simple case. From a mathematical point of view, I think I understand the idea, i.e., why it provides a valid probability measure for some measurement. What I don't understand is, how to construct the POVM for a given physical situation. You are right in demanding that one has to define, how the observed intensity pattern provides a weak common measurement of position and momentum of the incoming photon (which is difficult of course, because there's no position operator for photons to begin with, but here I understand it as the position on the screen where the photon was detected; you can of course also use neutrons or other massive particles, where this formal quibble doesn't occur).
This optical double-slit version of the Heisenberg microscope is, by the way, realized in a more sophisticated way with entangled photon pairs in Dopfner's PhD thesis. So it's simple but not completely academic:
http://people.isy.liu.se/jalar/kurser/QF/assignments/Dopfer1998.pdf
This optical double-slit version of the Heisenberg microscope is, by the way, realized in a more sophisticated way with entangled photon pairs in Dopfner's PhD thesis. So it's simple but not completely academic:
http://people.isy.liu.se/jalar/kurser/QF/assignments/Dopfer1998.pdf