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 ${\psi }_1^{\prime }+{\psi }_2^{\prime }$, which has the same numerical value as ${\psi }_1+{\psi }_2$, such that ${\psi }_1^{\prime }+{\psi }_2^{\prime }={\psi }_1+{\psi }_2$, but with ${\psi }_1^{\prime }=\frac{{\psi }_1+{\psi }_2}{\sqrt{2}}{e}^{ı\frac{\pi }{4}}\ne {\psi }_1$ and ${\psi }_2^{\prime }=\frac{{\psi }_1+{\psi }_2}{\sqrt{2}}{e}^{-ı\frac{\pi }{4}}\ne {\psi }_2$. Here ${\psi }_1^{\prime }$ and ${\psi }_2^{\prime }$ are the correct contributions from the slits to the total wave function ${\psi }_1^{\prime }+{\psi }_2^{\prime }$. We have then $p=|{\psi }_1^{\prime }+{\psi }_2^{\prime }{|}^2=|{\psi }_1^{\prime }{|}^2+|{\psi }_2^{\prime }{|}^2=p_1+p_2$ such that the paradox that quantum mechanics (QM) would not follow the traditional rules of probability calculus disappears.

also available on HAL at CEA-01459890v3