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Canadian Journal of Physics

Wave equation for dissipative systems derived from a quantized many-body problem

Mohsen Razavy

Published Jul 1, 1980• 7 citations
Publisher

Abstract

A classical many-body problem composed of an infinite number of mass points coupled together by springs is quantized. The masses and the spring constants in this system are chosen in such a way that the motion of each particle is exponentially damped. Because of the quadratic form of the Hamiltonian, the many-body wave function of the system can be written as a product of two terms: a time-dependent phase factor which contains correlations between the classical motions of the particles, and a stationary state solution of the Schrödinger equation. By assuming a Hartree type wave function for the many-particle Schrödinger equation, the contribution of the time-dependent part to the single particle wave function is determined, and it is shown that the time-dependent wave function of each mass point satisfies the nonlinear Schrödinger–Langevin equation. The characteristic decay time of any part of the subsystem, in this model, is related to the stiffness of the springs, and is the same for all particles.

DOI
10.1139/p80-140

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A classical many-body problem composed of an infinite number of mass points coupled together by springs is quantized. The masses and the spring constants in this system are chosen in such a way that the motion of each particle is exponentially damped. Because of the quadratic form of the Hamiltonian, the many-body wave function of the system can be written as a product of two terms: a time-dependent phase factor which contains correlations between the classical motions of the particles, and a stationary state solution of the Schrödinger equation. By assuming a Hartree type wave function for the many-particle Schrödinger equation, the contribution of the time-dependent part to the single particle wave function is determined, and it is shown that the time-dependent wave function of each mass point satisfies the nonlinear Schrödinger–Langevin equation. The characteristic decay time of any part of the subsystem, in this model, is related to the stiffness of the springs, and is the same for all particles.
From a quantized many-body system a wave equation for the motion of a particle linearly coupled to a heat bath is derived. The effective Hamiltonian describing the motion of the single particle is explicitly time dependent, and for a quadratic potential, has a simple dependence on the initial position and momentum of the particle. For the case of dissipative harmonic motion, a time-dependent wave equation is derived and the ground-state wave function is determined. It is also shown that if the equations of motion for the many-body system is Galilean invariant, the reduced form of equation of motion for the single particle is not. However a generalized form of transformation for the position and momentum operators, to a coordinate system moving with constant velocity, is obtained, which reduces to the Galilean transformation when the coupling to the dissipative system is turned off.
The close connection between the Jacobi function for a velocity-dependent dissipative system and the nonlinear Schrödinger–Langevin equation is used to study some of the features of the latter equation. Starting with the Hamiltonian for a classical system, different forms of the Jacobi generating function are obtained. With the help of these, corresponding solutions of the nonlinear wave equation are studied. It is found that the solution of the wave equation is not unique, and that it admits quasi-stationary, decaying, and asymptotically stationary solutions. Since this nonlinear wave equation is derivable from a classical generating function, it preserves some of the characteristic properties of the linear quantum theory. Among these are: (a) the lack of any correlation between noninteracting particles moving in a viscous medium, (b) the gauge independence of the observable quantities, and (c) the constancy of the density of states in phase space. This discussion can also be extended to the problem of quantal description of a harmonically bound radiating electron.
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.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.
We examine the two major approaches that have been suggested for the quantum mechanical treatment of the damped motion of a particle as a one-body problem. These are the linear, but time dependent, Kanai Hamiltonian, and the more recent nonlinear potentials which have been introduced to simulate the damping force. The most important criticism that has been leveled at the Kanai Hamiltonian is that its solutions seem to violate the uncertainty relations. We show that this Hamiltonian actually represents a particle of variable mass, whose classical behavior is identical to that of a damped particle of constant mass. But quantum mechanically, its changing mass does lead to unphysical behavior when misinterpreted as a constant mass particle. So this Hamiltonian cannot directly describe a constant mass damped quantum particle. The nonlinear model has been interpreted in terms of the hydrodynamical analogy of quantum theory, and a well behaved decaying wavepacket solution has been produced. However we generalize this result to produce solutions that ’’decay’’ to arbitrarily high energy. Thus it is not clear that this model specifically treats dissipation. Rather it seems to seek out any stationary state. At any rate, its physical interpretation is obscure at present. However we show, by analyzing the physical problem of damping at low energies, that one can modify the Kanai Hamiltonian to eliminate its unphysical features, so that this modified Kanai Hamiltonian can in fact be interpreted as representing a constant mass damped particle with physically reasonable solutions.
Phenomenological Equations of Motion for Dissipative Systems Lagrangian Hamiltonian and Hamilton-Jacobi Formulation of the Classical Dissipative Systems Noether's Theorem and Non-Noether Conservation Laws Dissipative Forces Derived from Many-Body Problems A Particle Coupled to a Field and the Damped Motion of a Central Particle Coupled to a Heat Bath Quantization of Dissipative Systems in General and of Explicitly Time-Dependent Hamiltonians in Particular Density Matrix and the Wigner Distribution Function for Damped Systems Path Integral Formulation of a Damped Harmonic Oscillator Quantization of the Motion of an Infinite Chain Heisenberg's Equations of Motion for a Particle Coupled to a Heat Bath Quantum Mechanical Models of Dissipative Systems and the Concept of Optical Potential.