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

Connection between quantum-mechanical and classical time evolution via a dynamical invariant

Dieter Schuch, M. Moshińsky

Published Jun 19, 2006• 31 citations
Publisher

Abstract

The time evolution of a quantum system with at most quadratic Hamiltonian is described with the help of different methods, namely the time-dependent Schr\"odinger equation, the time propagator or Feynman kernel, and the Wigner function. It is shown that all three methods are connected via a dynamical invariant, the so-called Ermakov invariant. This invariant introduces explicitly the quantum aspect via the position uncertainty and its possible time dependence. The importance of this aspect, also for the difference between classical and quantum dynamics, and in particular the role of the initial position uncertainty is investigated.

DOI
10.1103/physreva.73.062111

Related papers

Connection between quantum mechanical and classical time evolution of certain dissipative systems via a dynamical invariant

Dieter Schuch

In a former publication [D. Schuch and M. Moshinsky, Phys. Rev. A 73, 062111 (2006)] we have shown that the time evolution of a quantum system with at most quadratic Hamiltonian that can be described by different methods, namely, the time-dependent Schrödinger equation, the time propagator or Feynman kernel, and the Wigner function is connected via a dynamical invariant, the so-called Ermakov invariant. Since exact invariants of this type also exist for certain effective descriptions of dissipative quantum systems, we will show the relations between these descriptions and the corresponding invariants and will discuss how the results obtained for the conservative systems must be modified in the presence of a linear velocity dependent frictional force.

Journal of Mathematical Physics•Dec 1, 2007•22 citations

Riccati and Ermakov Equations in Time-Dependent and Time-Independent Quantum Systems

Dieter Schuch

The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system.The dynamics of the maximum, located at the mean value of position, is governed by the Newtonian equation of the corresponding classical problem.The width, which is directly proportional to the position uncertainty, obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinear Ermakov equation.The coupled pair of these equations yields a dynamical invariant which plays a key role in our investigation.It can be expressed in terms of a complex variable that linearizes the Riccati equation.This variable also provides the time-dependent parameters that characterize the Green's function, or Feynman kernel, of the corresponding problem.From there, also the relation between the classical and quantum dynamics of the systems can be obtained.Furthermore, the close connection between the Ermakov invariant and the Wigner function will be shown.Factorization of the dynamical invariant allows for comparison with creation/annihilation operators and supersymmetry where the partner potentials fulfil (real) Riccati equations.This provides the link to a nonlinear formulation of timeindependent quantum mechanics in terms of an Ermakov equation for the amplitude of the stationary state wave functions combined with a conservation law.Comparison with SUSY and the time-dependent problems concludes our analysis.

arXiv:0805.1687•Symmetry Integrability and Geometry Methods and Applications•May 12, 2008•24 citations

Relations of canonical and unitary transformations for a general time-dependent quadratic Hamiltonian system

Kyu-Hwang Yeon, Dukhyeon Kim, Chung‐In Um +2 more

We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classical equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The wave functions and the propagators are evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q and P, but the values of the functions of q and the kinetic momentum ${\mathrm{p}}_{\mathrm{k}}$ are equal to those of the other Q and ${\mathrm{P}}_{\mathrm{k}}$ in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.

Physical Review A•Jun 1, 1997•52 citations

Exact quantum theory of a time-dependent bound quadratic Hamiltonian system

Kyu Hwang Yeon, Kang Ku Lee, Chung‐In Um +2 more

Using the path-integral method, the propagator, the wave function, and the expectation values are evaluated explicitly for a time-dependent bound quadratic Hamiltonian system. We also have derived the relation between the wave function and a dynamical invariant which determines whether or not the system is bound. The expectation value of the quantum-mechanical invariant obeys the uncertainty relation with an auxiliary condition as the solution of the classical equation of the system.

Physical Review A•Oct 1, 1993•78 citations

Dynamical invariants in systems with and without broken time-reversal symmetry

Dieter Schuch

In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.

AIP conference proceedings•Jan 1, 2011•4 citations

Dynamical invariants in systems with and without broken time-reversal symmetry

Dieter Schuch, R. Bijker, Octavio Castaños +2 more

In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.

AIP conference proceedings•Jan 1, 2011•0 citations
The time evolution of a quantum system with at most quadratic Hamiltonian is described with the help of different methods, namely the time-dependent Schr"odinger equation, the time propagator or Feynman kernel, and the Wigner function. It is shown that all three methods are connected via a dynamical invariant, the so-called Ermakov invariant. This invariant introduces explicitly the quantum aspect via the position uncertainty and its possible time dependence. The importance of this aspect, also for the difference between classical and quantum dynamics, and in particular the role of the initial position uncertainty is investigated.
In a former publication [D. Schuch and M. Moshinsky, Phys. Rev. A 73, 062111 (2006)] we have shown that the time evolution of a quantum system with at most quadratic Hamiltonian that can be described by different methods, namely, the time-dependent Schrödinger equation, the time propagator or Feynman kernel, and the Wigner function is connected via a dynamical invariant, the so-called Ermakov invariant. Since exact invariants of this type also exist for certain effective descriptions of dissipative quantum systems, we will show the relations between these descriptions and the corresponding invariants and will discuss how the results obtained for the conservative systems must be modified in the presence of a linear velocity dependent frictional force.
The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system.The dynamics of the maximum, located at the mean value of position, is governed by the Newtonian equation of the corresponding classical problem.The width, which is directly proportional to the position uncertainty, obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinear Ermakov equation.The coupled pair of these equations yields a dynamical invariant which plays a key role in our investigation.It can be expressed in terms of a complex variable that linearizes the Riccati equation.This variable also provides the time-dependent parameters that characterize the Green's function, or Feynman kernel, of the corresponding problem.From there, also the relation between the classical and quantum dynamics of the systems can be obtained.Furthermore, the close connection between the Ermakov invariant and the Wigner function will be shown.Factorization of the dynamical invariant allows for comparison with creation/annihilation operators and supersymmetry where the partner potentials fulfil (real) Riccati equations.This provides the link to a nonlinear formulation of timeindependent quantum mechanics in terms of an Ermakov equation for the amplitude of the stationary state wave functions combined with a conservation law.Comparison with SUSY and the time-dependent problems concludes our analysis.
We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classical equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The wave functions and the propagators are evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q and P, but the values of the functions of q and the kinetic momentum pk{\mathrm{p}}_{\mathrm{k}}pk​ are equal to those of the other Q and Pk{\mathrm{P}}_{\mathrm{k}}Pk​ in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.
In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.
In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.