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Journal of Physics A Mathematical and General

Integro-differential Schrodinger equation for dissipative systems

H Cooper

Published Sep 7, 1993• 1 citation
Publisher

Abstract

The canonical treatment of the quantum dynamics of systems with dissipation affected degrees of freedom proceeds within the sum over histories picture of quantum theory. This paper proposes to supplement that approach with a Schrodinger equation approach to the same physics. Starting from a Lagrangian that is of the same form as those obtained by path integration, one can arrive at a quantized Hamiltonian analogue using the standard rules applied to non-dissipative systems. The resulting Hamiltonian is non-diagonal in position representation, so that the corresponding Schrodinger equation is a nonlinear integro-differential equation reminiscent of the Hartree equation. In the case of the normal tunnel junction, with a quasiparticle heat bath, this equation has an exact solution.

DOI
10.1088/0305-4470/26/17/002

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The canonical treatment of the quantum dynamics of systems with dissipation affected degrees of freedom proceeds within the sum over histories picture of quantum theory. This paper proposes to supplement that approach with a Schrodinger equation approach to the same physics. Starting from a Lagrangian that is of the same form as those obtained by path integration, one can arrive at a quantized Hamiltonian analogue using the standard rules applied to non-dissipative systems. The resulting Hamiltonian is non-diagonal in position representation, so that the corresponding Schrodinger equation is a nonlinear integro-differential equation reminiscent of the Hartree equation. In the case of the normal tunnel junction, with a quasiparticle heat bath, this equation has an exact solution.
The quantum-mechanical problem of tunneling of a particle coupled to a heat bath that consists of a large number of harmonic oscillators is studied in this paper. Here the equation of motion for the particle which is a nonlinear integro-differential equation is found by eliminating the degrees of freedom of the oscillators. The nonlinear operator equation may be solved by noting that the position operator at the time t can be expanded as a power series in terms of the initial position and momentum with time-dependent c-number coefficients. These coefficients also satisfy nonlinear integro-differential equations. For the problem of dissipative tunneling of a particle in a double-well potential, the first few terms of expansion have been calculated numerically. The expansion parameter in this formulation is a small dimensionless parameter which is inversely proportional to the product of the width and the square root of the height of the barrier.
The motion of a particle subject to an external force and coupled to a large number of harmonic oscillators is reducible to a nonlinear integro-differential equation for the position operator. This is achieved by eliminating the degrees of freedom corresponding to the motion of the oscillators. Upon this elimination the commutator of the position and the mechanical momentum of the particle becomes time dependent and a c number for a quadratic potential, but in general a q number for other fields of force. For a harmonically bound particle interacting with the heat bath the time evolution of the position and momentum operators are given in terms of the solutions of an integro-differential equation. Certain forms of the coupling of the motion of the particle to the heat bath allow for further reduction of these equations to quartic differential equations with constant coefficients. For these cases the time dependence of the position-momentum commutator, the energy of the particle as well as the heat bath are studied.
A wave equation formulation of the problem of quantum tunneling in a dissipative medium is developed by considering a many-body system in which the central particle is subject to an arbitrary force law, and at the same time is coupled to a bath of noninteracting harmonic oscillators. For the motion of the central particle it is possible to obtain an effective Lagrangian and Hamiltonian by eliminating the degrees of freedom of the oscillators. However both of these operators are nonlocal, and it is difficult to derive a wave equation for this motion. As an alternative method one can write a many-body Schrödinger equation for the whole system, and then eliminate the wave functions of all of the oscillators. This result is a many-channel Schrödinger equation for the motion of the central particle. By truncating this set of coupled equations, one can solve the problem for different force laws. In particular, in this work, the cases of dissipative tunneling, hopping, and quantum coherence are studied in detail. It is also shown how this approach can be generalized to multidimensional dissipative systems.
The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.
It has been shown that the irreversible dynamics of a system in a dissipative environment can be described by an effective one-particle Hamiltonian (in the classical case) or an effective one-particle Schrödinger equation (in the quantum mechanical case) without taking into account the environmental degrees of freedom explicitly. In the quantum mechanical case, the description can be achieved either by a nonlinear modification of the Schrödinger equation or an explicitly time-dependent Hamiltonian operator — where both forms are connected via a non-unitary transformation, corresponding to a non-canonical transformation on the classical level. Taking into account an external magnetic field, in both formalisms, an additional unphysical gauge-dependent term occurs in the equations of motion. This shortcoming can be eliminated if the Lagrangian and Hamiltonian that provide Maxwell’s equations of electrodynamics undergo, in the dissipative case, the same kind of transformation as the mechanical degrees of freedom of the system. As a result, a consistent description in terms of exact analytical wave packet-solutions for the dissipative motion in a magnetic field can be obtained that will be discussed in detail.
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.