Various classical limits, centre of mass and coherent states

Abstract

Two simple measuring processes with various classical limits are discussed using coherent states of the centre of mass ofN particles acting as a detector. the large-N and small-ħ limits are two typical ones. The small-ħ limit corresponds to a large external electric field or large-time limit in our models. It is shown that in the large-N limit all polynomials in averaged operators of the particles are classical, while in the small-ħ limit all polynomials in position and momentum operators of all particles are classical. In the latter case, the number of particlesN does not play an essential role at all, so that we can takeN=1. That is, it is possible to construct a detector consisting of a single particle. We also show that in the case ofN harmonic oscillators they will disturb the measuring process.