The Mott Problem in One Dimension

In the alpha decay of a nucleus, the tracks left in the medium by the alpha particle are linear, even though its initial wave function is spherically symmetric. Understanding this quantum phenomenon has been called "the Mott problem", ever since Mott's fundamental paper on the subject (Mott in Proc. R. Soc. London Ser. A 126:79 1929). Here we study a one dimensional version of the Mott problem. The particle emitted in the decay is represented as a superposition of waves, one traveling to the left, the other to the right. The atoms with which the particle interacts are modeled as two level systems. The wave equation obeyed by the particle is taken to be the massless Dirac equation. For a certain space-time structure for the particle-atom interaction, it is possible to derive an explicit space-time solution for the entire system, for an arbitrary number of atoms. In the one dimensional solution, the coherent superposition of right and left-moving wave packets leaves behind tracks of excited atoms. The Mott problem on the nature of the tracks left behind is addressed using the reduced density matrix, defined by taking the trace over all particle degrees of freedom. It is found that the reduced density matrix is the incoherent sum of two terms, one involving excited atoms only on the right; the other involving excited atoms only on the left, implying that tracks will show excited atoms on one side or the other. In one dimension, tracks which involve excited atoms exclusively on one side or the other are the analog of straight tracks in three dimensions.