A new theory for the description of dissipative systems by nonlinear Schrödinger-type field equations (NLSE’s) with logarithmic nonlinearity, which has been recently developed by the authors, is applied to investigate the frictionally damped free motion and similar spatially unrestricted aperiodic problems. Wave-packet solutions as well as time-dependent wave-function solutions are derived and discussed. In the limit of vanishing friction (friction constant γ→0) these solutions turn into the well-known solutions of the respective linear Schrödinger field equation. The same applies to the mean values of position, momentum, and energy, as well as to the uncertainty product of position and momentum. For γ≠0, however, interesting new effects appear. In contrast to the linear theory the uncertainty product of position and momentum does not diverge any more for infinitely long times, t→∞, but asymptotically approaches a definite constant value which depends on characteristic parameters of the system like its mass, initial width, and friction constant γ. Another effect, the faster spreading of the Gaussian wave-packet solution compared to the linear theory, can be explained with the help of a special property of our nonlinear differential equation. In a way similar to what is usually only known for linear differential equations, the wave-packet solutions of our NLSE can be obtained by superposition of the wave functions which are individually also solutions of the same NLSE. The properties of the time-dependent superposition coefficients appearing in this connection are discussed. The extension to the corresponding three-dimensional problem as well as the differences arising in the investigation of the NLSE’s of the free fall and the motion in a constant electric field are given. Concluding, some differences are discussed which appear applying our nonlinear field theory to describe periodic or aperiodic motions, respectively.
Abstract With 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)ω y, Ez = −(m/q)ω z, 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.
Taking into account wave-particle duality and the uncertainty principle, a nonlinear Schrödinger wave equation (NLSE) for the description of dissipative systems can be constructed by applying only Newton's form of classical mechanics. A comparison between our NLSE for dissipative systems and the linear time-dependent SE for conservative systems shows that, for free motion and motion in a harmonic potential, both equations have Gaussian wave-packet solutions. The time dependence of the maximum of these functions is governed by the classical (real) Newtonian equation of motion. While the time dependence of the width of the wave packet can be determined from a nonlinear complex Bernoulli/Ricatti type differential equation. This can be done by transformation into a Newton-type equation for a real quantity α (which is proportional to the square root of the width), including a kind of “centrifugal force” term. This equation can be solved analytically and when taken with Newton's equation of motion for the mean value of the position, it forms a so-called Ermakov system. For the linear SE equation, this implies the existence of a dynamical invariant. For our NLSE we also obtain such an exact invariant, but now for a dissipative system. Furthermore, this complex nonlinear equation can be linearised using the logarithmic derivative of a complex quantity λ, thus yielding a complex Newtonian equation of motion. The absolute value of λ is equivalent to α, the phase angle can be determined using a relation that connects real and imaginary parts of λ. This relation is linked to a kind of “conservation of angular momentum” for the quantity λ in a complex plane. The relations between the above-mentioned quantities will be discussed in detail.
The quantum Brownian motion of a particle in an external potential V(x) is treated using the master equation for the Wigner distribution function W(x, p, t) in phase space (x, p). A heuristic method of determination of diffusion coefficients in the master equation is proposed. The time evolution equation so obtained contains explicit quantum correction terms up to o(4) and in the classical limit, → 0, reduces to the Klein–Kramers equation. For a quantum oscillator, the method yields an evolution equation for W(x, p, t) coinciding with that of Agarwal (1971 Phys. Rev. A 4 739). In the non-inertial regime, by applying the Brinkman expansion of the momentum distribution in Weber functions (Brinkman 1956 Physica 22 29), the corresponding semiclassical Smoluchowski equation is derived.