The consequences of measurement in quantum mechanics I. An idealized trajectory determination

Abstract

An idealized thought experiment is presented which illustrates certain features of the theory of measurement in quantum mechanics. Measurements of momentum and time-of-passage of a particle are proposed which provide information on the previous history of the particle insofar as it can be described by an equation of the form x = x 0 + v 0 t. Particular features of the measurement process that are illustrated are (a) simultaneous measurements of time and energy, limited in precision by a Heisenberg relation; (b) the selection of a portion of the incoming wave-packet by magnetic deflection and a slit system; (c) the approach with increasing resolution of any one of the selected portions towards an eigenstate of momentum; and (d) the irrelevance in this case of Von Neumann's projection postulate and the “reduction of the wave packet” phenomenon because the approach to an eigenstate occurs before the measurement is completed. It is assumed that the operation of a detector aperture can be described in terms of a weight function multiplying the wavefunction in momentum space, the absolute square of which gives the detector efficiency function. Successive measurements are not treated in this paper, which follows the usual quantum-mechanical practice of dealing with the unfolding-in-time of states only between a preparation and a single act of measurement. Three kinds of uncertainty principle are distinguished, dealing, respectively, with independent, simultaneous, and successive measurements.