The Mott problem concerns the paradox of reconciling the spherical wave function describing the emission of an alpha ray by a radioactive nucleus, with the linear tracks seen in a cloud chamber.
Formulated in 1927 by Albert Einstein and Max Born[citation needed], it was resolved by a calculation done by Sir Nevill Francis Mott that showed that the correct quantum mechanical system must include the wave functions for the atoms in the cloud chamber as well as that for the alpha ray.
The calculation showed that the resulting probability is non-zero only on straight lines raying out from the decayed atom; that is, once the measurement is performed, the wave-function becomes non-vanishing only near the classical trajectory of a particle.
In Renninger's 1960 formulation, the cloud chamber is replaced by a pair of hemispherical particle detectors, completely surrounding a radioactive atom at the center that is about to decay by emitting an alpha ray.
The full collapse of the wave function, down to a single point, does not occur until it interacts with the outer hemisphere.
The radii are chosen so that the more distant hemisphere is much farther away than the half-life of the decaying nucleus, times the flight-time of the alpha ray.
Another common objection states that the decay particle was always travelling in a straight line, and that only the probability of the distribution is spherical.
The distinction between mixed and pure states is illustrated more clearly in a different context, in the debate comparing the ideas behind local-hidden variables and their refutation by means of the Bell inequalities.
If a diffraction pattern were not observed, one would be forced to conclude that the particle had collapsed down to a ray, and stayed that way, as it passed the inner hemisphere; this is clearly at odds with standard quantum mechanics.
Even if the initial state could be polarized; for example, by placing it in a magnetic field, the non-spherical decay pattern is still properly described by quantum mechanics.
The above formulation is inherently phrased in a non-relativistic language; and it is noted that elementary particles have relativistic decay products.