The Brownian motion of a particle coupled to a heat bath can be derived from an exactly solvable many-body system. This motion is governed by the Langevin equation in Newtonian mechanics, and by an operator equation formally identical to the Langevin equation when the system composed of the particle plus the heat bath is quantized. To get this formal similarity between the classical and the quantal equations of motion of the particle, the classical dissipative system is formulated in terms of a modified Hamilton–Jacobi equation and is then quantized using the Schrödinger method. The result is identical with the Schrödinger–Langevin equation that has been obtained by quantizing the entire system and then isolating the motion of the particle. The non-linear wave equation describing the motion of a particle subject to conservative and time-dependent forces as well as frictional forces has been applied to the problems of motion of a wave-packet, and of the scattering and trapping of heavy-ions.
The physical model of a nonrelativistic quantized Schrodinger's electron (SE) is offered. The behaviour of the SE well spread elementary electric charge had been understood by means of two independent and different in a frequency and size motions. The description of this resultant motion may be done by substitution of the classical Wiener continuous integral with the quantized Feynmam continuous integral. There are possibility to show by means of the existent not only formal but substantial analogy between the quadratic differential wave equation in partial derivatives of Schrodinger and quadratic differential particle equation in partial derivatives of Hamilton-Jacoby that the addition of a kinetic energy of the stochastic harmonic oscillation of some quantized micro particles to the kinetic energy of classical motion of the same micro particles formally determines their wave behaviour.It turns out the stochastic motion of the quantized micro particles powerfully to break up the smooth thin line of the classical motion of the same micro particle in many broad cylindrically spread path. The SE participate in stochastically roughly determined circumferences within different flats and with different radii, with centres which are successively arranjed over short and very disorderly orientated lines. Therefore the quantized motion of some micro particle cannot be descripted by smooth thin well contured (focused) line, describing the classical motion of the macro particle.
The orbit equation for the motion of a quantum in a potential is proposed by defining uniquely the quantum mechanical momentum from the Schrodinger wave equation of a one-particle system and using the Hamilton-Jacobi formalism. On this occasion it is necessary to introduce a new dynamical postulate, “initial condition”, which is independent of the quantization condition as well as of the Schrodinger wave equation. The momentum is determined in such a way that it is approximately equal to the corresponding classical mechanical momentum in the classical region of validity of the coordinates. The momentum in the non-classical region is in general very small in comparison with that in the classical region. The quantum-mechanical motion well corresponds to the classical motion. If a system is macroscopic, the motion of the particle is described by Newtonian mechanics and the quantization condition is of no validity. The “quantization of magnetic flux” derives from the quantization condition. It is closely related to the energy level structure of the atomic system. The above argument is applied to the scalar electromagnetic and sound wave equations to give the dynamical motion of a photon and a phonon without recourse to the so-called second quantization method.