Connection between the Schrödinger equation for dissipative systems and the master equation

1982 ◽  
Vol 116 (1-2) ◽  
pp. 248-264 ◽  
Author(s):  
P. Caldirola ◽  
L.A. Lugiato
Keyword(s):  
Schrödinger Equation ◽  
Master Equation ◽  
Schrodinger Equation ◽  
Dissipative Systems

scholarly journals Spatiotemporal Complexity Mediated by Higher-Order Peregrine-Like Extreme Events

2021 ◽  
Vol 9 ◽  
Author(s):  
Saliya Coulibaly ◽  
Camus G. L. Tiofack ◽  
Marcel G. Clerc
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Analytical Description ◽  
Rogue Waves ◽  
Dissipative Systems ◽  
Largest Lyapunov Exponent ◽  
Dissipative Effects ◽  
Non Linear ◽  
The Stability ◽  
Good Agreement

The Peregrine soliton is the famous coherent solution of the non-linear Schrödinger equation, which presents many of the characteristics of rogue waves. Usually studied in conservative systems, when dissipative effects of injection and loss of energy are included, these intrigued waves can disappear. If they are preserved, their role in the dynamics is unknown. Here, we consider this solution in the framework of dissipative systems. Using the paradigmatic model of the driven and damped non-linear Schrödinger equation, the profile of a stationary Peregrine-type solution has been found. Hence, the Peregrine soliton waves are persistent in systems outside of the equilibrium. In the weak dissipative limit, analytical description has a good agreement with the numerical simulations. The stability has been studied numerically. The large bursts that emerge from the instability are analyzed by means of the local largest Lyapunov exponent. The observed spatiotemporal complexity is ruled by the unstable second-order Peregrine-type soliton.

scholarly journals Solution of the Master Equation for Quantum Brownian Motion Given by the Schrödinger Equation

Mathematics ◽  
2016 ◽  
Vol 5 (1) ◽  
pp. 1
Author(s):  
R. Sinuvasan ◽  
Andronikos Paliathanasis ◽  
Richard Morris ◽  
Peter Leach
Keyword(s):  
Brownian Motion ◽  
Schrödinger Equation ◽  
Master Equation ◽  
Schrodinger Equation ◽  
Quantum Brownian Motion

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

1980 ◽  
Vol 58 (7) ◽  
pp. 1019-1025 ◽  
Author(s):  
M. Razavy
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Body Problem ◽  
Body Wave ◽  
Time Dependent ◽  
Dissipative Systems ◽  
Many Body ◽  
Dependent Part ◽  
The Many

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.

Quantum tunneling in a dissipative environment

1992 ◽  
Vol 70 (9) ◽  
pp. 719-730 ◽  
Author(s):  
M. Hron ◽  
M. Razavy
Keyword(s):  
Wave Equation ◽  
Schrödinger Equation ◽  
Quantum Tunneling ◽  
Schrodinger Equation ◽  
Quantum Coherence ◽  
Harmonic Oscillators ◽  
Dissipative Systems ◽  
Many Body ◽  
Coupled Equations ◽  
Central Particle

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.

Sign in / Sign up

Export Citation Format

Share Document