Finite-time measurement of quantum particle’s mean position

We analyze nonrelativistic quantum measurement of the time average of the particle’s coordinate, $X\equiv t^{-1}{\int }_0^t{x(t}^{\prime }{)dt}^{\prime }.$ The measurement amplitude is constructed by restricting the Feynman path integral to paths with the required value of X. The resulting decomposition of the Schrödinger wave function determines the type of meter needed to measure X. We show that such meter can be realized as a magnetic moment traveling with the particle in a magnetic field whose magnitude linearly changes with x. Weak and strong measurement regimes are discussed.