A predictive solution of the EPR paradox
Henryk Gzyl
Abstract
In this work we solve the apparent paradox proposed by EPR. The paradox appears when the momentum of a particle is known with certainty without measuring it, and its coordinate can be measured with arbitrary precision, thus contradicting the uncertainty principle. To prove that there is no paradox we can go two ways. First, we prove that if one measures the total momentum of the system and then predicts the momentum of one particle, this automatically yields a prediction of the momentum of the second. If after first measuring the total momentum, one then measures the momentum of, say, the first particle, that automatically yields the momentum of the second. The issue is that in both cases, the momentum of the second, depends on the total momentum of the system, which does not commute with the position of the second particle, and no contradiction to the uncertainty principle happens. We verify these assertions by computing the relevant quantum conditional expectations, and we also establish that this is equivalent to computing the expected values with respect to the von Neumann post-measurement state.