Zitterbewegung in Statistical Systems with Space Dependent Potentials?

  In a number of notes , a linear 2x2 matrix Dirac type equation was examined as a solution of Einstein’s 1905 energy momentum equation E2=p2+m2 (1). This model was able to reproduce Schrodinger zitterbewegung.To do this, however, one had to identify the 2x2 energy matrix H as being part of the evolution operator exp(-iHt). It was seen that the identity matrix together with H could transform any two vector into the eigenvector of H, which is taken as the stable solution. This idea of an evolution operator thus seems to be important.

  In addition, the model led to a 2-eigenvector which was interpreted as the square root of two components of probability, one in which a particle moves forward the other in which it moves backward Overall, the particle moves forward with a velocity of v along the x axis. Knuth has also proposed such a model. It was found that the probabilities which came from this quantum mechanical type matrix solution matched those obtained by maximizing  statistical entropy. In a note , the 2x2 matrix model was applied to a system of classical particles which could have energy e1 or e2. It was argued that the 2x2 model predicted zitterbewegung for this statistical system. The object of this note is to explore whether the 2x2 model can be used to predict zitterbewegung in a statistical system with a potential.