# Probability calculus and Quantum Mechanics [closed]

I am not an expert and probably this question highlights this. Anyway, is the probability calculus used in Quantum Mechanics? Does the concept of probability adopted in Quantum Mechanics satisfy the rules of probability calculus?

• Yes, it is used. In the context of measurement all relevant observables commute, so the resulting probability space is classical. One can go beyond that and define quantum probability, which is non-classical, but this formalism is not mandatory, and standard expositions do not use it. Apr 22 '19 at 5:48
• @Conifold what does it mean that “observables commute”? Apr 22 '19 at 15:17
• Observables are measurable physical quantities, in quantum mechanics they are represented by self-adjoint operators. When these operators commute the corresponding observables can be measured simultaneously. Apr 23 '19 at 16:24

The famous 1964 paper of John Stewart Bell, in which "Bell's inequality" is established, begins by assuming two things. (The paper can be found here: https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf)

One of them is "locality", which means (non-mathematically) that distant objects cannot affect one another instantaneously. Mathematically, Bell assumes that the function $A$ cannot depend on $\vec{b}$, and the function $B$ cannot depend on $\vec{a}$.

The other assumption is that there are "hidden variables". What matters to your question is that Bell's formulation of this assumption is mathematically the same as assuming that the experiment can be modeled on a probability space. The variable $\lambda$ in Bell's paper is nothing other than an outcome in a sample space.

Bell then goes on to derive a contradiction. What we can conclude is that, under the assumption of locality, there are quantum mechanical models that cannot be expressed using a probability space, and are therefore inconsistent with probability theory.

Probabilities in quantum mechanics always respect the calculus of probabilities by definition. The square amplitudes of a set of orthogonal states, which are the quantities used to compute probabilities in quantum mechanics, don't always respect the calculus of probability, see:

https://arxiv.org/abs/math/9911150

Probability is the connection between the mathematical apparatus of quantum mechanics and experimental observation, data in the real world (it's what gives quantum mechanics the status of a scientific theory). A more appropriate question (for the philosophy section ) is related to the multitude of interpretations of quantum mechanics, but there is not enough room on the whole stackexchange to do that subject any justice.