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

Journal of Physics A Mathematical and General

Gauge-covariant properties of a linear nonautonomous quantum system: time-dependent even and odd coherent states

L. F. Wei, Shun-Jin Wang, X. L. Lei

Published Jan 7, 2002• 7 citations
Publisher

Abstract

A new method for finding the dynamical invariant for a general time-dependent harmonic oscillator is proposed by making use of two linearly independent solutions to the classical equation of motion. It is shown that the dynamical invariant for different gauged Hamiltonians are connected by time-dependent gauge transformations. Therefore, the representation whose bases are the instantaneous eigenstates of the invariant operator is a good representation for the quantum nonautonomous system. In this representation, the wavefunction of the system is gauge covariant and thus any observable physical effect is naturally independent of the choice of gauged Hamiltonians. The exact even and odd coherent states for a time-dependent harmonic oscillator are constructed in terms of these gauge covariant bases. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.

DOI
10.1088/0305-4470/35/2/319

Related papers

Even and Odd Coherent States for Time-Dependent Harmonic Oscillator

Lian-Fu Wei, Qingyi Yang, Shun-Jin Wang

The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time-dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.

Communications in Theoretical Physics•Jul 15, 2002•1 citation

Gauge transformation approach to the exact solution of a generalized harmonic oscillator

Fuli Li, Shun-Jin Wang, A. Weiguny +1 more

The exact solution and the invariant operator of a generalized harmonic oscillator are obtained by performing three consecutive gauge transformations on the time-dependent Schrodinger equation. In contrast to previous results, they depend only on solutions of a linear differential equation for the associated classical harmonic oscillator. Non-adiabatic and adiabatic Berry phases for the system are calculated on the basis of the exact solution.

Journal of Physics A Mathematical and General•Feb 7, 1994•39 citations

Quantum Action-Angle Variables for the Harmonic Oscillator

Harry R. Lewis, W. E. Lawrence, Joseph David Harris

Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of $q$ and the eigenstates of $p$. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.

Physical Review Letters•Dec 23, 1996•27 citations

An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field

Harry R. Lewis, W. B. Riesenfeld

The theory of explicitly time-dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well-posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time-dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time-dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time-dependent uniform charge distribution. A class of explicitly time-dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.

Journal of Mathematical Physics•Aug 1, 1969•1,800 citations

Exact evolution of the generalized damped harmonic oscillator

Mustapha Maamache, Hichem Choutri

In this paper we use the algebraic and the invariant method to study the time-dependent damped harmonic oscillator from classical and quantum points of view. The solution of the classical equation of motion and the wavefunction solving the time-dependent Schrödinger equation are found explicitly. We show that the original time-dependent quantum-mechanical problem is completely related to the well known time-independent harmonic oscillator. In addition, we elucidate the intimate connection between the damped harmonic oscillator (DHO) and the generalized harmonic oscillator (GHO). More importantly, the evolution of the states of the DHO cannot be cyclic, in contradistinction with the states of the GHO. Explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit. The coherent states describing the invariant-angle variables of the classical DHO are constructed; they allow us to recover the classical evolution and invariant from the quantum evolution.

Journal of Physics A Mathematical and General•Aug 25, 2000•25 citations

COMPLETE EXACT QUANTUM STATES OF THE GENERALIZED TIME-DEPENDENT HARMONIC OSCILLATOR

I. A. Pedrosa

By making use of linear and quadratic invariants and the invariant operator formulation of Lewis and Riesenfeld, the complete exact solutions of the Schrödinger equation for the generalized time-dependent harmonic oscillator are obtained. It is shown that the general solution of the system under consideration contains both the discrete and continuous spectrum. The connection between linear and quadratic invariants and their corresponding eigenstates via time-dependent auxiliary equations is also established.

Modern Physics Letters B•Oct 20, 2004•14 citations
A new method for finding the dynamical invariant for a general time-dependent harmonic oscillator is proposed by making use of two linearly independent solutions to the classical equation of motion. It is shown that the dynamical invariant for different gauged Hamiltonians are connected by time-dependent gauge transformations. Therefore, the representation whose bases are the instantaneous eigenstates of the invariant operator is a good representation for the quantum nonautonomous system. In this representation, the wavefunction of the system is gauge covariant and thus any observable physical effect is naturally independent of the choice of gauged Hamiltonians. The exact even and odd coherent states for a time-dependent harmonic oscillator are constructed in terms of these gauge covariant bases. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.
The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time-dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.
Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of qqq and the eigenstates of ppp. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.
The theory of explicitly time-dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well-posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time-dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time-dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time-dependent uniform charge distribution. A class of explicitly time-dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.
In this paper we use the algebraic and the invariant method to study the time-dependent damped harmonic oscillator from classical and quantum points of view. The solution of the classical equation of motion and the wavefunction solving the time-dependent Schrödinger equation are found explicitly. We show that the original time-dependent quantum-mechanical problem is completely related to the well known time-independent harmonic oscillator. In addition, we elucidate the intimate connection between the damped harmonic oscillator (DHO) and the generalized harmonic oscillator (GHO). More importantly, the evolution of the states of the DHO cannot be cyclic, in contradistinction with the states of the GHO. Explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit. The coherent states describing the invariant-angle variables of the classical DHO are constructed; they allow us to recover the classical evolution and invariant from the quantum evolution.
By making use of linear and quadratic invariants and the invariant operator formulation of Lewis and Riesenfeld, the complete exact solutions of the Schrödinger equation for the generalized time-dependent harmonic oscillator are obtained. It is shown that the general solution of the system under consideration contains both the discrete and continuous spectrum. The connection between linear and quadratic invariants and their corresponding eigenstates via time-dependent auxiliary equations is also established.