A Model of the Cloud Chamber in Generalized Coherent State Formalism

Abstract

Particle trajectory in the Wilson cloud chamber is explained by means of the perturbation theory and generalized coherent states. The model consists of an incoming particle, molecules and detectors (the immediate environment of the molecules). The particle and the molecules are treated as typical quantum systems, while each detector has many harmonic oscillators whose state are totally de­ scribed by generalized coherent states. If a molecule is ionized through interaction with a particle, the ionized excited state is stable, that is, the molecule does not return to the ground The stability of excited states of the molecule is a result of properties of the generalized coherent states of the detector interacting with the molecule. § 1. Inh:oduction In a previous paper l ) we have presented two models for the quantum measuring process with the help of generalized coherent (GC) states, and shown that the detectors in these models have many desirable functions as measuring apparatus. These models also indicate that the GC state formalism can become a powerful method for a unified description of various measuring processes.· The GC state formalism is based on a group theoretical method developed by many authors.2) Many quantum theories have some measure of the number of dynamical variables N (or lin); for a wide class of these quantum theories the large-N limit is known to simplify the dynamics dramatically. In these theories the vacuum expectation of any product of reasonable operators satisfies the factorization relation, that is, the expec­ tation approaches the product of each expectation of the operators in the large-N limit. As a result, fluctuations of these operators become irrelevant when N is large. Also in some sense quantum theories with large N behave like classical theories with a phase space, the Poisson bracket, and classical Hamiltoni~n. Coherence group and GC states play an important role in this group theoretical method. The GC state approach to quantum measurement may provide a link between the microscopic and macroscopic descriptions of detectors. The essential features of the Wilson cloud chamber have already been explained in two ways using the perturbation theory. The two ways come from the arbitrari­ ness in the concept of observation, i.e., whether the molecules ·to be ionized are regarded as belonging to the observed system or the observing apparatus. These treatments of the cloud chamber are justified from Copenhagen's point of view. 3 )-5) The difference between the above two ways will disappear in GC state formalism, because the molecules can only be the observed system, not the observing apparatus. An observing apparatus must, in our formalism, have GC states generated by applying elements of a coherence group to a base state. In this article, we will apply the GC state formalism to the above cloud chamber. An incoming particle and molecules are typical quantum systems; the environment of