About conditional probabilities of events regarding the quantum mechanical measurement process

We consider the successive measurement of position and momentum of a single particle. Let P be the conditional probability to measure the momentum k with precision Δk, given a previously successful position measurement q with precision Δq. Several upper bounds for the probability P are derived. For arbitrary, but given precisions Δq and Δk, these bounds refer to the variation of q, k, and the state vector ψ of the particle. The first bound is given by the inequality P ≤ ΔkΔqh, where h is Planck’s quantum of action. It is nontrivial for all measurements with ΔkΔq < h. A sharper bound is obtained by applying the Hilbert-Schmidt norm. As our main result, the least upper bound of P is determined. All bounds are independent of the order with which the measuring of the position and momentum is made