Mott problem
The Mott problem is an iconic challenge to quantum mechanics theory: how can the prediction of spherically symmetric wave function result in linear tracks seen in a cloud chamber.
[1]: 119ff The problem was first formulated in 1927 by Albert Einstein and Max Born and solved in 1929 by Nevill Francis Mott.
[2] Mott's solution notably only uses the wave equation, not wavefunction collapse, and it is considered the earliest example of what is now called decoherence theory.
[3] The problem later associated with Mott concerns a spherical wave function associated with an alpha ray emitted from the decay of a radioactive atomic nucleus.
[3] Intuitively, one might think that such a wave function should randomly ionize atoms throughout the cloud chamber, but this is not the case.
The result of such a decay is always observed as linear tracks seen in Wilson's cloud chamber.
In practice, virtually all high energy physics experiments, such as those conducted at particle colliders, involve wave functions which are inherently spherical.
Yet, when the results of a particle collision are detected, they are invariably in the form of linear tracks (see, for example, the illustrations accompanying the article on bubble chambers).
It is somewhat strange to think that a spherically symmetric wave function should be observed as a straight track, and yet, this occurs on a daily basis in all particle collider experiments.
[4]: 160 Max Born described the problem as one that Albert Einstein pointed to, asking "how can the corpuscular character of the phenomenon be reconciled here with the representation by waves?".
Born answers with Heisenberg's "reduction of the probability packet", now called wavefunction collapse, introduced in May 1927.
[3]: 220 In his highly influential 1930 book,[5] Werner Heisenberg analyzed the problem qualitatively but in detail.
He says the correct approach requires viewing the wavefunction as consisting of the system under study (the alpha particle) and the environment it interacts with (atoms of the cloud chamber).
Starting with a simple spherical wave, each collision involves a wavefunction with more coordinates and increasing complexity.
) are fixed during the calculation of the track, meaning the velocity of the alpha particle is taken as much larger than the thermal motion of the gas atoms.
These relative coordinates are parameters in the solution so the intensity of the excitations for various positions can be compared.
By assuming that the emitter and the hydrogen atoms are not close together, Mott represents the time-independent part of the three-body state of the system,
[3] Consequently, Mott is assuming that the alpha particle barely notices the atoms it excites as it races through the cloud chamber.
which describes the scattered alpha-particle wave when the first atom is excited and the second is in its ground state.
[2][3] Mott demonstrated that by considering the interaction in configuration space, where all of the atoms of the cloud chamber play a role, it is overwhelmingly probable that all of the condensed droplets in the cloud chamber will lie close to the same straight line.
[7] Mott's analysis, while it predates modern decoherence theory, fits squarely within its approach.
