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You can find derivation of these operators in most standard quantum mechanics textbooks. For your convenience, see https://en.wikipedia.org/wiki/Momentum_operator and https://en.wikipedia.org/wiki/Position_operator. For the second question, Paul Dirac said in his classic The Principles of Quantum Mechanics: A measurement always causes the system to jump ...

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The former and the latter are really the same: "$c_n=\psi(x)$". If you want to measure positions, then possible outcome states are $|x\rangle$, therefore you write $$|\psi\rangle = \sum_x|x\rangle\langle x|\psi\rangle:=\sum_x\psi(x)|x\rangle :=\sum_xc_x|x\rangle$$ This tells you, the probability to find the particle at position $x$, i.e. to measure it in ...

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You have fallen prey to the same confusion that many people have with regards to the wave/particle duality: The quantum objects that constitute our world are neither waves nor classical particles, and it is an error to believe that electrons/photons/whatever can "propagate as a wave" in one moment and "behave like a particle" in the next. The quantum ...

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