Wave-function collapse by measurement and its simulation.

Abstract

The many-Hilbert-space approach to the measurement problem in quantum mechanics is applied to a typical ``yes-no'' experiment relative to two branch routes corresponding to mutually exclusive propositions. First, we reformulate the notion of wave-function collapse by measurement as a dephasing process between the two branch waves of an interfering particle (from our own point of view as opposed to the conventional Copenhagen interpretation). In this way, the concept of ``wave-function collapse'' is replaced by that of a statistically defined dephasing process. One of the most important points of this paper is the introduction of an order parameter \ensuremath{\epsilon} that quantitatively describes the degree of decoherence. Its value ranges from \ensuremath{\epsilon}=0 (which describes the case in which the two waves are perfectly coherent) to \ensuremath{\epsilon}=1 (which describes the case in which coherence is totally lost); for this reason \ensuremath{\epsilon} is named the ``decoherence parameter.'' In terms of this parameter we formulate a definite criterion to judge whether an instrument works well or not as a measuring apparatus. Then, we study the interaction between a microscopic particle and a macroscopic system (a detector), by modeling the macrosystem with a linear array of complex \ensuremath{\delta} potentials, which undergo several kinds of statistical fluctuations. This leads us, under particular conditions, to the so-called wave-function collapse, which is attained in the limit \ensuremath{\epsilon}=1. We also examine in some detail which kind of elastic and/or inelastic collisions can give the wave-function collapse. Some connections with recent experimental results in neutron interferometry and quantum optics are also stressed.