arXiv:0712.4083 · Physica Scripta

The damped Pinney equation and its applications to dissipative quantum mechanics

Fernando Haas

Published Jan 15, 2010• 22 citations

Abstract

The present work considers the damped Pinney equation, defined as the model arising when a linear in velocity damping term is included in the Pinney equation. In the general case, the resulting equation does not admit Lie point symmetries or reduction to a simpler form by any obvious coordinate transformation. In this context, the method of Kuzmak–Luke is applied to derive a perturbation solution, for weak damping and slow time dependence of the frequency function. The perturbative and numerical solutions are shown to be in good agreement. The results are applied to examine the time evolution of Gaussian-shaped wave functions in the Kostin formulation of dissipative quantum mechanics.

DOI: 10.1088/0031-8949/81/02/025004

Related papers

Time-Dependent Gaussian Solution for the Kostin Equation around Classical Trajectories

Fernando Haas, José Maria Filardo Bassalo, Antonio Boulhosa Nassar +1 more

The structure of time-dependent Gaussian solutions for the Kostin equation in dissipative quantum mechanics is analyzed. Expanding the generic external potential near the center of mass of the wave packet, one conclude that: the center of mass follows the dynamics of a classical particle under the external potential and a damping proportional to the velocity; the width of the wave packet satisfy a non-conservative Pinney equation. An appropriate perturbation theory is developed for the free particle case, solving the long standing problem of finding analytic expressions for square integrable solutions of the free Kostin equation. The associated Wigner function is also studied.

Aug 14, 20166 citations

Models of damped oscillators in quantum mechanics

Ricardo Erwin Suazo

We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.

Journal of Physical MathematicsJan 1, 20096 citations

Models of damped oscillators in quantum mechanics

Ricardo Cordero–Soto, Erwin Suazo, Sergeĭ K. Suslov

arXiv:0905.0507Journal of Physical MathematicsJan 1, 200940 citations

Dissipative quantum dynamics: solution of the generalized von Neumann equation for the damped harmonic oscillator

Martin Hensel, H. J. Korsch

A recently proposed generalized von Neumann equation describing dissipative time evolution of quantum systems is applied to a damped and driven harmonic oscillator. Utilizing Lie algebraic methods the nonlinear operator equation is solved and from this solution the behaviour of the mean value (x) is extracted, including both amplitude and phaseshift relative to the driving force. A differential equation for (x) is derived and the damping constant is identified.

Journal of Physics A Mathematical and GeneralApr 7, 199214 citations

Dissipation in Quantum Mechanics. The Harmonic Oscillator

I. R. Senitzky

The need for a quantum-mechanical formalism for systems with dissipation which is applicable to the radiation field of a cavity is discussed. Two methods that have been used in this connection are described. The first, which starts with the classical Newtonian equation of motion for a damped oscillator and applies the conventional formal quantization techniques, leads to an exact solution; but subsequent discussion shows that this method is invalid, the results being unacceptable from a quantum-mechanical viewpoint. The second method, which considers the interaction of two systems, the lossless oscillator and the loss mechanism, is adopted in the present article. No special model is used for the loss mechanism, but this mechanism is assumed to have a large number of densely-spaced energy states.The approximations with respect to the loss mechanism that underlie the concept of dissipation are discussed. These approximations are then applied to the analysis, and a differential equation for a coordinate operator of the harmonic oscillator is obtained which has the formal appearance of the Newtonian equation of motion for a driven damped harmonic oscillator, the driving term being an operator referring to the loss mechanism. The presence of the driving term is responsible for the difference between the present theory and that of the first method mentioned above. A solution of the differential equation for the coordinate operator is given explicitly. An examination of the physical significance of the solution shows that the driving term is responsible not only for the thermal fluctuations which are due to the loss mechanism, but also for the proper commutation relationship of the conjugate coordinates of the oscillator and for its zero-point fluctuations.A generalization of the solution to provide for a classical driving force and coupled atomic systems is given. The results are then restated in a form that refers to the loss mechanism only through the two parameters by which it is usually described---the dissipation constant and the temperature.

Physical ReviewJul 15, 1960466 citations

Path integral formulation of dissipative quantum dynamics

Alexey Novikov

In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

Jan 20, 20050 citations