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International Journal of Quantum Chemistry

Quantum‐classical transition of the dissipative wave packet dynamics for barrier scattering

Chia‐Chun Chou

Published Oct 9, 2018• 5 citations
Publisher

Abstract

Abstract A nonlinear equation based on the Schrödinger–Langevin equation is employed to analyze the quantum‐classical transition of dissipative systems for wave packet barrier scattering. The transition equation is obtained by subtracting a modified quantum potential term depending on a degree of quantumness in the Schrödinger–Langevin equation, and it provides a continuous description for the transition process of dissipative systems from purely quantum to purely classical regimes. Important properties possessed by the equation are derived. The energy dissipation rate of the dissipative system does not depend on the degree of quantumness. It is shown that the equations of motion for the expectation values of the position and momentum operators for dissipative systems hold for all the degree of quantumness. This feature establishes the correspondence between classical and quantum dissipative systems. Computational results for the transition process of dissipative wave packet dynamics and transition trajectories are presented and analyzed for model systems involving the propagation of a free Gaussian wave packet and the wave packet scattering from an Eckart barrier. Computational results demonstrate that the agreement between the classical‐limit transition trajectories and classical trajectories is excellent, except in some regions. The single‐valued transition wave function implies that transition trajectories cannot pass through the same configuration space point at the same time.

DOI
10.1002/qua.25812

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Iranian Journal of Physics Research•May 1, 2021•0 citations
Abstract A nonlinear equation based on the Schrödinger–Langevin equation is employed to analyze the quantum‐classical transition of dissipative systems for wave packet barrier scattering. The transition equation is obtained by subtracting a modified quantum potential term depending on a degree of quantumness in the Schrödinger–Langevin equation, and it provides a continuous description for the transition process of dissipative systems from purely quantum to purely classical regimes. Important properties possessed by the equation are derived. The energy dissipation rate of the dissipative system does not depend on the degree of quantumness. It is shown that the equations of motion for the expectation values of the position and momentum operators for dissipative systems hold for all the degree of quantumness. This feature establishes the correspondence between classical and quantum dissipative systems. Computational results for the transition process of dissipative wave packet dynamics and transition trajectories are presented and analyzed for model systems involving the propagation of a free Gaussian wave packet and the wave packet scattering from an Eckart barrier. Computational results demonstrate that the agreement between the classical‐limit transition trajectories and classical trajectories is excellent, except in some regions. The single‐valued transition wave function implies that transition trajectories cannot pass through the same configuration space point at the same time.
Abstract A quantum‐classical transition equation in complex space is derived in the framework of the complex quantum Hamilton–Jacobi formalism. The transition equation is obtained by subtracting a complex‐valued quantum potential term from the complex‐extended time‐dependent Schrödinger equation (TDSE). It is shown that the nonlinear transition equation is equivalent to a linear scaled TDSE with a rescaled Planck's constant. Employing the quantum momentum function defined by the gradient of the complex action, we can analyze the quantum‐classical transition of physical systems using complex transition trajectories. Complex transition trajectories are presented for the free Gaussian wave packet, the harmonic oscillator, and the Morse potential. This study demonstrates that the transition equation provides a continuous description for the quantum‐classical transition of physical systems in complex space.
A nonlinear quantum–classical transition wave equation is proposed for dissipative systems within the Caldirola-Kanai model. Equivalence of this transition equation to a scaled Schrödinger equation is proved. The dissipative dynamics is then studied in terms of what we call scaled trajectories following the standard procedure used in Bohmian mechanics. These trajectories depend on a continuous parameter allowing us a smooth transition from Bohmian to classical trajectories. Arrival times and actual momentum distribution functions are also analyzed. The propagation of a Gaussian wave packet in a viscid medium under the presence of constant, linear and harmonic potentials is studied. The gradual decoherence process and localization are easily visualized and understood within this theoretical framework.
The Caldeira-Leggett master equation, expressed in Lindblad form, has been used in the numerical study of the effect of a thermal environment on the dynamics of the scattering of a wave packet from a repulsive Eckart barrier. The dynamics are studied in terms of phase space trajectories associated with the distribution function, W(q,p,t). The equations of motion for the trajectories include quantum terms that introduce nonlocality into the motion, which imply that an ensemble of correlated trajectories needs to be propagated. However, use of the derivative propagation method (DPM) allows each trajectory to be propagated individually. This is achieved by deriving equations of motion for the partial derivatives of W(q,p,t) that appear in the master equation. The effects of dissipation on the trajectories are studied and results are shown for the transmission probability. On short time scales, decoherence is demonstrated by a swelling of trajectories into momentum space. For a nondissipative system, a comparison is made of the DPM with the "exact" transmission probability calculated from a fixed grid calculation.
Abstract The free propagation in time of a normalisable wave packet is the oldest problem of continuum quantum mechanics. Its motion from microscopic to macroscopic distance is the way in which most quantum systems are detected experimentally. Although much studied and analysed since 1927 and presented in many textbooks, here the problem is re‐appraised from the standpoint of semi‐classical mechanics. Particular aspects are the emergence of deterministic trajectories of particles emanating from a region of atomic dimension and the interpretation of the wave function as describing a single particle or an ensemble of identical particles. Of possible wave packets, that of Gaussian form is most studied due to the simple exact form of the time‐dependent solution in real and in momentum space. Furthermore, this form is important in laser optics. Here the equivalence of the time‐dependent Schrödinger equation to the paraxial equation for the propagation of light is demonstrated explicitly. This parallel helps to understand the relevance of trajectory concepts and the conditions necessary for the perception of quantum motion as classical. Key Points The paper addresses a problem of fundamental quantum mechanics, important in both physics and chemistry. Within physics, the subject is relevant to the disciplines of both quantum mechanics and classical optics. The matter presented is topical also for researchers in the history and philosophy of physics.
Quantum systems interacting with a dissipative environment can effectively be described by a nonlinear modification of the Schrodinger equation. For some standard problems that possess analytical solutions, e.g. the free motion and the harmonic oscillator, also the nonlinear equation can be solved in a closed form and has Gaussian-shaped wave packet solutions. The equation determining the time-evolution of the wave-packet width is - already in the case of the usual conservative Schrodinger equation - nonlinear and provides the quantities that are necessary to construct the Green’s function for the respective system. The Green’s functions for the conservative and dissipative cases will be given explicitly in position and momentum space.
Ignoring thermal fluctuations of the environment, taking into account only its dissipative effects, free propagation of a Gaussian wavepacket is studied in the framework of the linear Caldirola-Kanai (CK) equation and non-linear equations, the Schrodinger-Langevin (SL) equation, known as Kostin equation, and the Schuch-Chung-Hartmann (SCH) equation. By a Gaussian ansatz for the probability density one obtains two equations, one for the evolution of the center of the wave packet which is just the classical Langeving equation and one for the evolution of the width of the wavepacket. This last equation has different forms in different approaches having only an analytic solution in the CK and SCH frameworks. Computations show that for a given friction, the width of the wavepacket increases with time in all approaches. In a given time, it reduces with friction in both CK and SL approaches revealing localization effects of dissipation, while has opposite behavior in the SCH approach. Furthermore, energy expectation value and its time-derivative is computed and compared in all approaches. It is shown that the rate of energy is given by the expectation value of momentum field in the SL framework. Finally, transmission of a Gaussian wavepacket from a rectangular barrier is numerically studied in the context of CK and compared to the non-dissipative case. The transmission decreases considerably with the dissipation.