QUANTUM DENSITY OF PROBABILITY AT THE CLASSICAL PECULIAR POINT

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 po st-Galilean approximation only.(1-4) In this approximation the Lagrang ians are singular on some surfaces of the phase space. The dynamical p roperties are essentially peculiar on the singular surfaces.(5-8) In t he particular case of the rectilinear motion of two electrons the pecu liar point appears when the distance between the particles r = r(0), w here r(0) = e(2)/mc(2) (the so-called ''radius of an electron''). Here m and e are respectively the mass and the charge of the electron, c i s the speed of light. In this paper it is shown that in the simple cas e of a one-dimensional system of two electrons with the symmetrical in itial condition nu(1) = -nu(2) (nu(1) and nu(2) are the velocities of the particles), the density of probability tends to zero when the dist ance between electrons tends to r(0). In other words, the point of the classical phase-space, which cannot be crossed by the trajectory of t he system, reflects at the point where the corresponding quantum syste m has the vanishing probability.