A solution of the paradox of the double-slit experiment
Résumé
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.
The paradox is rooted in the wrong intuition that $\psi_{1}$ and $\psi_{2}$ would be the true physical contributions to
$\psi'_{1} + \psi'_{2} =\psi_{1} + \psi_{2}$ like in the case of waves in a water tank.
The solution proposed here is not ${ad {hoc}}$ but based on an extensive analysis of the geometrical meaning of spinors within group representation theory and its application to QM. Working further on the argument one can even show that an interference pattern is the only way to satisfy simultaneously two conditions: The condition obeying binary logic (in the spirit of Einstein) that the electron has only two mutually exclusive options to get to the detector (viz. going through slit S$_{1}$ or going through slit S$_{2}$) and the condition obeying ternary logic (in the spirit of Bohr) that the question which one of these two options the electron has taken is experimentally undecidable.
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