On the statistical derivation of the Schrödinger equation

Abstract

The transition from reversible microscopic operator equations to irreversible equations for a deterministic density matrix is considered for examples of “simple systems”—the hydrogen atom or a free electron in an electromagnetic field. As a result of the transition, a system of a particle and field oscillators is replaced by a continuous medium. The Schrödinger equation for the deterministic wave function also describes the evolution of a continuum but without allowance for dissipative terms. In this sense, there is an analogy between the Schrödinger equation in quantum mechanics and Euler's equation in hydrodynamics. The smallest size of a “point” of a continuous medium is described by the classical electron radiusr _e . It also determines the effective Thomson cross section for scattering of photons by free electrons. The lengthr _e and the corresponding time interval τ _e =r _e /c play the role of “hidden parameters” in quantum mechanics. Two methods of calculating the effective Thomson cross section in terms of the extinction coefficient are considered. The first of them is based on the equation of motion of a free electron in a field with allowance for radiative friction. This equation leads to well-known difficulties. Moreover, the velocity fluctuations calculated on its basis lead to a contradiction with the second law of thermodynamics. The second method is based on the introduction of a constant friction coefficient Ń = τ ⁻¹_e , the presence of which reflects loss of information on smoothing over the volume of a “point” of the continuous medium. Such a method of calculation leads to the same expression for the effective cross section but makes it possible to avoid the difficulties with the second law of thermodynamics. In the derivation of quantum kinetic equations, the physically infinitesimally small scales are determined by the Compton length λ_C and the corresponding time interval. The introduction of these scales makes it possible to separate and eliminate small-scale fluctuations, the “collision integrals” being expressed in terms of the correlation functions of these fluctuations.