On Tracks in a Cloud Chamber

Abstract

It is an experimental fact that \(\alpha \)-decays produce in a cloud chamber at most one track (sequence of liquid droplets) and that this track points in a random direction. This seems to contradict the description of decay in Quantum Mechanics: according to Gamow a spherical wave is produced and moves radially according to Schrodinger’s equation. It is as if the interaction with the supersaturated vapor turned the wave into a particle. The aim of this note is to place this effect in the context of Schrodinger’s Quantum Mechanics. We shall see that the properties of the initial wave function suggest the introduction of a semiclassical formalism in which the \(\alpha \)-wave can be described as a collection of semiclassical (probability) wavelets; each of them interacts with an atom and forms an entangled state. The interaction can be regarded as a semiclassical inelastic scattering event. The measurement (of the position of the first droplet of the track) selects the wave function of one of the ions, with a probability given by Born’s rule. This ion interacts with the atoms nearby leading to the formation of a droplet. One can reasonably assume that also the wavelet entangled with the selected ion has probability one to remain as part of the description of the system. The measurement process is therefore represented by a (non local) unitary operator. The (semiclassical) wavelet remains sharply localized on a classical path \( \gamma _{class} .\) It is still a probability wave: it determines the probability that another atom be ionized. This probability is essentially zero unless the atom lies on \( \gamma _{class}\). This gives the visible track.