Some remarks on classical probability theory in quantum mechanics

Abstract

Essentially this note is a comment to a paper which has been withdrawn by the author. I would hope that it could simplify the discussion about prob? ability theory in quantum mechanics. The slit experiment is the classical example of demonstrating the non classical behaviour of quantum mechanical probability theory. But to get this demonstration, it is necessary that we pay attention to important experimental conditions. First of all, it is important that we prepare the beam of electrons or the light beam in the right way so that an interference pattern can be seen. Secondly, we have to perform a whole class of experi? ments to demonstrate the desired nonclassical behaviour of quantum me? chanical probability theory. For instance, we have to open and shut one of the two slits. It is not only a two-slit experiment as was already remarked by C. Piron in his paper. For more details see for example the article by P. Mittelstaedt [1]. But if the beam has been prepared in the wrong way it is, nevertheless, possible to see no interference effects, and then we can give a classical probability description of the whole class of experiments. This is the case if a bad physicist is performing the experiment. Therefore, it is not very surprising that many typical quantum mechanical experiments like for instance the radioactive decay disregarding some aspects, for example spin and momenta have a classical probability description. Perhaps a more general remark is necessary concerning probabilistic descriptions of physical situations. If in a model all possible events have probabilities unequal to zero, such a probability distribution can never be wrong, but it can be very bad in practical applications. Therefore physicists often speak of a "wrong" (or an impossible probabilistic) description of a special physical situation, if they can give a probabilistic description, which is in a sense better than the "wrong" one. In this sense a probabilistic description of the interference experiment is not possible with the classical probability theory, because the classical theory does not describe the "cor? rect" interference terms. In the same sense you should read the non-ex? istence proofs of hidden variable theories, such as I had given in my book