Emergence of classical trajectories in quantum systems: the cloud chamber problem in the analysis of Mott (1929)

We analyze the paper “The wave mechanics of $\alpha$-ray tracks” (Mott, 1929), published in 1929 by N.F. Mott. In particular, we discuss the theoretical context in which the paper appeared and give a detailed account of the approach used by the author and the main result attained. Moreover, we comment on the relevance of the work not only as far as foundations of Quantum Mechanics are concerned but also as the earliest pioneering contribution in decoherence theory.

💡 Research Summary

In his 1929 paper “The wave mechanics of α‑ray tracks,” N. F. Mott tackled the long‑standing puzzle of why a quantum particle, emitted as a spherical wave, leaves a straight‑line track in a cloud chamber. At the time, quantum mechanics had embraced wave‑particle duality, yet it offered no clear mechanism for the emergence of classical trajectories from the underlying wave function. Mott’s approach was to treat the whole system—an α particle and the surrounding gas atoms—as a single multi‑particle quantum system and to analyse the full wave function ψ(r,R₁,…,R_N) that depends on the coordinates of the α particle (r) and of each of the N atoms (R_i).

He began by writing the time‑independent Schrödinger equation for this composite system:

(H₀ + ∑_{i=1}^{N} V_i) ψ = E ψ,

where H₀ contains the kinetic energies of the free α particle and the free atoms, and V_i represents the short‑range interaction potential between the α particle and the i‑th atom. The crucial assumption is that each V_i is appreciable only when the α particle passes very close to the atom, allowing a perturbative treatment of each scattering event.

Mott first considered the scattering off a single atom. The α particle is initially described by a spherical outgoing wave e^{ik·r}/r. After interacting with the first atom, the wave function can be expanded in spherical harmonics Y_{l m}(θ,φ) multiplied by radial functions. The l = 0 component (the isotropic part) is largely suppressed, while higher‑l components survive, producing a wave that is no longer isotropic but instead concentrated around a particular direction (θ₀, φ₀). In other words, the first scattering “collimates” the wave into a narrow cone.

The second step is to compute the transition amplitude for the α particle to ionise a second atom. This amplitude is ⟨ψ₁|V₂|ψ₁⟩, where ψ₁ is the post‑first‑scattering wave. Because ψ₁ already has most of its intensity in the cone defined by (θ₀, φ₀), the matrix element is appreciable only for atoms located within that cone. Atoms outside receive essentially zero probability. Consequently, the probability that the α particle will cause a second ionisation is strongly biased toward the same direction as the first ionisation.

Repeating the argument for the third, fourth, …, N‑th atoms leads to a recursive amplification: each successive scattering further narrows the angular distribution. Mathematically, the probability of n successive ionisations along the same line scales as

P_n ∝ |f(θ₀, φ₀)|^{2n},

where f(θ, φ) is the angular part of the wave function after the first scattering. The exponential dependence means that, even though the initial spherical wave gives an equal a priori chance to all directions, the chain of scatterings selects a single direction with overwhelming likelihood. The result is a series of ionisation events that line up almost perfectly, reproducing the straight tracks observed in cloud chambers.

Mott emphasized that the total wave function remains a pure state throughout; no external observer is required to “collapse” the wave function into a trajectory. The environment (the gas atoms) itself, through entanglement and successive scattering, dynamically selects a set of quasi‑classical “pointer states”—the straight‑line tracks. This insight anticipates the modern theory of decoherence, where the system’s interaction with many uncontrolled degrees of freedom suppresses interference between macroscopically distinct alternatives and stabilises particular classical outcomes.

Two key approximations underlie the analysis. First, the atom‑α interaction is short‑ranged, allowing the use of free‑particle wave functions with small perturbative corrections. Second, the atoms are assumed to be randomly, uniformly distributed in space, so that statistical averaging over their positions yields the same directional amplification for any initial emission direction. Under these conditions, the multi‑particle wave function exhibits a built‑in bias toward linear ionisation chains.

Mott’s final conclusion is that the classical picture of a particle moving along a definite trajectory emerges naturally from the quantum mechanical description once the collective effect of many weak, short‑range scatterings is taken into account. The paper thus provides the first quantitative, fully quantum‑mechanical explanation of cloud‑chamber tracks and, retrospectively, can be read as the earliest concrete example of decoherence in action. It bridges the gap between the abstract wave function and the concrete, observable world of classical particle paths, laying groundwork that would later be formalized in the decoherence program and influencing foundational discussions on quantum measurement.