A solution for the paradox of the double-slit experiment

Abstract: We argue that the double-slit experiment can be understood much better by considering it as an experiment whereby one uses electrons to study the set-up rather than an experiment whereby we use a set-up to study the behaviour of electrons. We also show how the concept of undecidability can be used in an intuitive way to make sense of the double-slit experiment and the quantum rules for calculating coherent and incoherent probabilities. We meet here a situation where the electrons always behave in a fully deterministic way (following Einstein's conception of reality), while the detailed design of the set-up may render the question about the way they move through the set-up experimentally undecidable (which follows more Bohr's conception of reality). We show that the expression $\psi_{1} + \psi_{2}$ for the wave function of the double-slit experiment is numerically correct, but logically flawed. It has to be replaced in the interference region by the logically correct expression $\psi'{1} + \psi'{2}$, which has the same numerical value as $\psi_{1} + \psi_{2}$, such that $\psi'{1} + \psi'{2} = \psi_{1} + \psi_{2}$, but with $\psi'{1} = {\psi{1} +\psi_{2}\over{\sqrt{2}}} \,e^{\imath {\pi\over{4}}} \neq \psi_{1}$ and $\psi'{2} = {\psi{1} +\psi_{2}\over{\sqrt{2}}}\,e^{-\imath {\pi\over{4}}}\neq \psi_{2}$. Here $\psi'{1}$ and $\psi'{2}$ are the correct contributions from the slits to the total wave function $\psi'{1} + \psi'{2}$. We have then $p = |\psi'{1} + \psi'{2}|^{2} = |\psi'{1}|^{2} + |\psi'{2}|^{2} = p_{1}+p_{2} $ such that the paradox that quantum mechanics (QM) would not follow the traditional rules of probability calculus disappears.