QUANTUM DENSITY OF PROBABILITY AT THE CLASSICAL PECULIAR POINT

Abstract

The so-called no-interaction theorem of D.G. Currie, T.F. Jordan, E.C. Sudarshan, H. Leutwyler, G. Marmo and N. Mukunda makes it possible to construct relativistic quasi-classical particle dynamics in the post-Galilean approximation only.1−4 In this approximation the Lagrangians are singular on some surfaces of the phase space. The dynamical properties are essentially peculiar on the singular surfaces.5−8 In the particular case of the rectilinear motion of two electrons the peculiar point appears when the distance between the particles r=r0, where r0=e2/mc2 (the so-called “radius of an electron”). Here m and e are respectively the mass and the charge of the electron, c is the speed of light. In this paper it is shown that in the simple case of a one-dimensional system of two electrons with the symmetrical initial condition v1=−v2 (v1 and v2 are the velocities of the particles), the density of probability tends to zero when the distance between electrons tends to r0. In other words, the point of the classical phase-space, which cannot be crossed by the trajectory of the system, reflects at the point where the corresponding quantum system has the vanishing probability.