Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Schrödinger equation for general linear velocity-dependent forces

Luana J. Faria Gonçalves, L. S. F. Olavo

Published Feb 2, 2018• 3 citations
Publisher

Abstract

Quantization processes generally assume the Hamiltonian formulation of classical mechanics. The notion of dissipation does not fit easily into the Hamiltonian structure. There appeared in the literature a quantization process that circumvents the use of the Hamiltonian approach and derives the Schr\"odinger equation from first principles. Thus, the usual approach of assuming a dissipative force of the type ${f}_{d}=k(t)v$, where $k(t)$ is a scalar depending only on the time and $v$ is the velocity, can be approached using this quantization process to mathematically derive a Schr\"odinger equation. The derivation for this simple case was already performed by the authors elsewhere. The full generalization of a dissipative force in the context of a linear response theory, as is usual in classical mechanics, would be to write this force as ${f}_{i}(\mathbf{x},t)={\mathcal{K}}_{ij}(\mathbf{x},t){v}_{j}$, where $\mathcal{K}(\mathbf{x},t)$ is a tensor giving the nonhomogeneity and nonisotropy of the process. In this paper, we present this generalization and connect our results with nonlinear Schr\"odinger-like equations already known from the literature.

DOI
10.1103/physreva.97.022102

Related papers

On the quantum mechanical treatment of dissipative systems

Rainer W. Hasse

Two types of Hamiltonians are investigated which describe quantum mechanically a particle moving subject to a linear viscous force under the influence of a conservative force: the conventional explicitly time-dependent one and an alternative class of nonlinear Hamiltonians. In the latter group we propose a new form. By Ehrenfest’s theorem the expectation values of the operators of physical observables correspond to the classical quantities. For all Schrödinger equations we derive and discuss wavepacket, wave, stationary, and pseudostationary solutions of force free motion, free fall, and harmonic oscillator.

Journal of Mathematical Physics•Oct 1, 1975•310 citations

Quantum Dynamics Using The Time-Dependent Schrödinger Equation

Abraham Nitzan

This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.

Oxford University Press eBooks•Apr 6, 2006•0 citations

Nonlinear wave equation for quantized frictional forces

Mohsen Razavy

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.

Canadian Journal of Physics•Oct 1, 1978•13 citations

Deformation quantization of linear dissipative systems

V. G. Kupriyanov, S. L. Lyakhovich, A. A. Sharapov

A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product $*_t$ and an ``extended'' operator of time derivative $D_t=\\partial_t+...$, differentiating the $\\ast_t$-product. As in the usual case, the $\\ast_t$-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional $\\mathrm{Tr}_t$. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.

•Jan 1, 2005•7 citations

Study about Forced Harmonic Oscillator in the Space (x, v)

Gustavo V. López, Omar J. P. Bravo

The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable (\(\mathit{x}\) , \(\hat{v}\)) , with the commutation relation [\(\mathit{x}\) , \(\hat{v}\)] = \(\mathit{i}\hbar\)/\(\mathit{m}\) , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion \(\mathit{K}\)(\(\mathit{x, v, t}\)) of the classical system, The comparison with the quantization in the space (\(\mathit{x, p}\)) is done with the usual Schrödinger’s equation for the Hamiltonian \(\mathit{H}\)(\(\mathit{x, p, t}\)), and with the commutation relation [\(\mathit{x}\) , \(\hat{p}\)] = \(\mathit{i}\hbar\). It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (\(\mathit{x}\) , \(\hat{v}\)) quantization has fewer oscillations than the (\(\mathit{x}\) , \(\hat{p}\)) quantization, the average energy of the system is higher in (\(\mathit{x}\) , \(\hat{p}\)) quantization than on the (\(\mathit{x}\) , \(\hat{v}\)) quantization, and the Boltzmann- Shannon entropy on the (\(\mathit{x}\) , \(\hat{p}\)) quantization is higher than on the (\(\mathit{x}\) , \(\hat{v}\)) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.

Book Publisher International (a part of SCIENCEDOMAIN International)•Sep 1, 2023•1 citation

Classical and Quantum Dissipative Systems

Mohsen Razavy

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.

WORLD SCIENTIFIC eBooks•Dec 13, 2016•85 citations
Quantization processes generally assume the Hamiltonian formulation of classical mechanics. The notion of dissipation does not fit easily into the Hamiltonian structure. There appeared in the literature a quantization process that circumvents the use of the Hamiltonian approach and derives the Schr"odinger equation from first principles. Thus, the usual approach of assuming a dissipative force of the type fd=k(t)v{f}_{d}=k(t)vfd​=k(t)v, where k(t)k(t)k(t) is a scalar depending only on the time and vvv is the velocity, can be approached using this quantization process to mathematically derive a Schr"odinger equation. The derivation for this simple case was already performed by the authors elsewhere. The full generalization of a dissipative force in the context of a linear response theory, as is usual in classical mechanics, would be to write this force as fi(x,t)=Kij(x,t)vj{f}_{i}(\mathbf{x},t)={\mathcal{K}}_{ij}(\mathbf{x},t){v}_{j}fi​(x,t)=Kij​(x,t)vj​, where K(x,t)\mathcal{K}(\mathbf{x},t)K(x,t) is a tensor giving the nonhomogeneity and nonisotropy of the process. In this paper, we present this generalization and connect our results with nonlinear Schr"odinger-like equations already known from the literature.
Two types of Hamiltonians are investigated which describe quantum mechanically a particle moving subject to a linear viscous force under the influence of a conservative force: the conventional explicitly time-dependent one and an alternative class of nonlinear Hamiltonians. In the latter group we propose a new form. By Ehrenfest’s theorem the expectation values of the operators of physical observables correspond to the classical quantities. For all Schrödinger equations we derive and discuss wavepacket, wave, stationary, and pseudostationary solutions of force free motion, free fall, and harmonic oscillator.
This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.
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.
A simple pseudo-Hamiltonian formulation is proposed for the linear inhomogeneous systems of ODEs. In contrast to the usual Hamiltonian mechanics, our approach is based on the use of non-stationary Poisson brackets, i.e. corresponding Poisson tensor is allowed to explicitly depend on time. Starting from this pseudo-Hamiltonian formulation we develop a consistent deformation quantization procedure involving a non-stationary star-product ∗t*_t∗t​ and an ``extended'' operator of time derivative Dt=partialt+...D_t=\\partial_t+...Dt​=partialt​+..., differentiating the astt\\ast_tastt​-product. As in the usual case, the astt\\ast_tastt​-algebra of physical observables is shown to admit an essentially unique (time dependent) trace functional mathrmTrt\\mathrm{Tr}_tmathrmTrt​. Using these ingredients we construct a complete and fully consistent quantum-mechanical description for any linear dynamical system with or without dissipation. The general quantization method is exemplified by the models of damped oscillator and radiating point charge.
The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable (x\mathit{x}x , v^\hat{v}v^) , with the commutation relation [x\mathit{x}x , v^\hat{v}v^] = iℏ\mathit{i}\hbariℏ/m\mathit{m}m , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion K\mathit{K}K(x,v,t\mathit{x, v, t}x,v,t) of the classical system, The comparison with the quantization in the space (x,p\mathit{x, p}x,p) is done with the usual Schrödinger’s equation for the Hamiltonian H\mathit{H}H(x,p,t\mathit{x, p, t}x,p,t), and with the commutation relation [x\mathit{x}x , p^\hat{p}p^​] = iℏ\mathit{i}\hbariℏ. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (x\mathit{x}x , v^\hat{v}v^) quantization has fewer oscillations than the (x\mathit{x}x , p^\hat{p}p^​) quantization, the average energy of the system is higher in (x\mathit{x}x , p^\hat{p}p^​) quantization than on the (x\mathit{x}x , v^\hat{v}v^) quantization, and the Boltzmann- Shannon entropy on the (x\mathit{x}x , p^\hat{p}p^​) quantization is higher than on the (x\mathit{x}x , v^\hat{v}v^) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.
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.