Nonlinear Schrödinger‐type field equation for the description of dissipative systems. II. Frictionally damped motion in a magnetic field

Abstract

AbstractWith the aid of a method similar to the one we used in an earlier work (part I) a new Schrödinger‐type field equation with logarithmic nonlinearity can be derived from a Fokker–Planck equation for a distribution function. This nonlinear field equation describes the frictinally damped motion of a system under the influence of a magnetic field and can be interpreted in the same way as the nonlinear Schrödinger‐type equation (NLSE) derived in part I, where no magnetic fields were taken into account. The NLSE for the two‐dimensional motion of a charged material system in a homogeneous magnetic field is solved exactly. The solutions are compared with the quantum‐mechanical solutions of the corresponding undamped problem. The method is extended to include also anisotropic conditions; i.e., in the Fokker–Plank equation the diffusion constant has to be replaced by a diffusion matrix, as different diffusion constants may be possible for different space directions. We regard the three‐dimensional motion under the combined influence of magnetic and electric fields according to K = (q/c)(v × B) + qE − mγv with Ey = (m/q)ωy, Ez = −(m/q)ωz, B = (0, 0, B) as an example. This is an approximation of the conditions existing in an ion cyclotron resonance spectrometry cell, neglecting an additional drift motion in the x direction which could be taken into account by Galilean transformation and gauge transformation. The corresponding NLSE for the coupled three‐dimensional motion is specified and solved exactly.