## 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 Schrödinger’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 Schrödinger’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.

## Keywords

Tracks in a cloud chamber Entanglement Semiclassical Position measurement## Notes

### Acknowledgments

Part of the contents of this note were presented in a stimulating Conference in Bielefeld on the foundations of Quantum Mechanics. I am grateful to Philippe Blanchard and Jurg Froelich for the invitation and for several inspiring discussions. I am grateful also for remarks and comments by my friends Rodolfo Figari and Sandro Teta with whom I discussed at length the role of Mott’s problem in Quantum Mechanics. I am also very grateful to an anonymous referee for pointing out several inaccuracies in the first draft; the comments have helped much to improve the presentation. I want to thank another referee for drawing my attention to reference [3].

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