Wigner transform and quantum-like corrections for charged-particle beam   transport
  It is shown that charged-particle beam transport in the paraxial approximation can be effectively described with a quantum-like picture in semiclassical approximation. In particular, the classical Liouville equation can be suitably replaced by a von Neuman-like equation. Relevant remarks concerning the standard classical description of the beam transport are given. 
R. Fedele, V.I. Man'ko