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

Feynman propagator for time-dependent Lagrangians possessing an invariant quadratic in momentum

Asish Kumar Dhara, S. V. Lawande

Published Aug 21, 1984• 42 citations
Publisher

Abstract

A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out an explicit path integration, that the propagator for such a time-dependent system is related to the propagator of an associated time-independent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of Lewis and Reisenfeld (1969) is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach.

DOI
10.1088/0305-4470/17/12/014

Related papers

Time-dependent invariants and the Feynman propagator

Asish Kumar Dhara, S. V. Lawande

A class of recently discussed time-dependent classical Lagrangians possessing invariants is considered from a quantum-mechanical point of view. Quantum mechanics is introduced directly through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out explicit path integration, that the propagator for these time-dependent problems is related to the propagator of associated time-independent problems. The expansion of the propagator in terms of the eigenfunctions of the invariant operator and the quantum superposition principle follow naturally in our scheme. The theory is applied to obtain explicitly exact propagators for some illustrative examples.

Physical Review A•Jul 1, 1984•45 citations

Time-dependent invariants and quantum mechanics in two dimensions

Asish Kumar Dhara, S. V. Lawande

A class of time-dependent problems in two space dimensions possessing time-dependent invariants, bilinear in momenta, is considered. Explicit expression of the potentials and the corresponding invariants are derived. Quantum mechanics is introduced in these time-dependent problems directly through a Feynman propagator defined as a path integral involving the classical action. The propagators are shown to admit expansions in terms of the eigenfunctions of the corresponding invariant operators. Equivalence of the present theory to that of Lewis and Riesenfeld [H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)] is discussed.

Journal of Mathematical Physics•May 1, 1986•4 citations

Adelic Path Integrals for Quadratic Lagrangians

Goran S. Djordjević, Branko G Dragovich, Ljubiša Nešić

Feynman's path integral in adelic quantum mechanics is considered. The propagator [Formula: see text] for one-dimensional adelic systems with quadratic Lagrangians is analytically evaluated. Obtained exact general formula has the form which is invariant under interchange of the number fields ℝ and ℚp.

arXiv:hep-th/0105030•Infinite Dimensional Analysis Quantum Probability and Related Topics•Jun 1, 2003•27 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

Exact solution of a time-dependent quantal harmonic oscillator with damping and a perturbative force

Dinkar C. Khandekar, S. V. Lawande

The problem of a quantal harmonic oscillator with damping and a time-dependent frequency acted on by a time-dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time-dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.

Journal of Mathematical Physics•Sep 1, 1979•85 citations

Path Integrals in Noncommutative Quantum Mechanics

Dragovich, B, Rakic, Z

Extension of Feynman's path integral to quantum mechanics of noncommuting spatial coordinates is considered. The corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians) is formulated. Our approach is based on the fact that a quantum-mechanical system with a noncommutative configuration space may be regarded as another effective system with commuting spatial coordinates. Since path integral for quadratic Lagrangians is exactly solvable and a general formula for probability amplitude exists, we restricted our research to this class of Lagrangians. We found general relation between quadratic Lagrangians in their commutative and noncommutative regimes. The corresponding noncommutative path integral is presented. This method is illustrated by two quantum-mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.

•Sep 22, 2003•20 citations
A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out an explicit path integration, that the propagator for such a time-dependent system is related to the propagator of an associated time-independent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of Lewis and Reisenfeld (1969) is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach.
A class of recently discussed time-dependent classical Lagrangians possessing invariants is considered from a quantum-mechanical point of view. Quantum mechanics is introduced directly through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out explicit path integration, that the propagator for these time-dependent problems is related to the propagator of associated time-independent problems. The expansion of the propagator in terms of the eigenfunctions of the invariant operator and the quantum superposition principle follow naturally in our scheme. The theory is applied to obtain explicitly exact propagators for some illustrative examples.
A class of time-dependent problems in two space dimensions possessing time-dependent invariants, bilinear in momenta, is considered. Explicit expression of the potentials and the corresponding invariants are derived. Quantum mechanics is introduced in these time-dependent problems directly through a Feynman propagator defined as a path integral involving the classical action. The propagators are shown to admit expansions in terms of the eigenfunctions of the corresponding invariant operators. Equivalence of the present theory to that of Lewis and Riesenfeld [H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)] is discussed.
The problem of a quantal harmonic oscillator with damping and a time-dependent frequency acted on by a time-dependent perturbative force is exactly solved. The wavefunctions are found in Schrödinger representation using the theory of explicitly time-dependent invariants and also by an expansion of the Feynman propagator. The propagator is obtained in exactly closed form by an explicit path integration of the classical Lagrangian. It is found that the wavefunctions and the propagator depend only on the solution of classical damped oscillator through a single function ρ (t). The function ρ (t) itself may be obtained as a solution of a second order nonlinear differential equation under the appropriate set of initial conditions.
Extension of Feynman's path integral to quantum mechanics of noncommuting spatial coordinates is considered. The corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians) is formulated. Our approach is based on the fact that a quantum-mechanical system with a noncommutative configuration space may be regarded as another effective system with commuting spatial coordinates. Since path integral for quadratic Lagrangians is exactly solvable and a general formula for probability amplitude exists, we restricted our research to this class of Lagrangians. We found general relation between quadratic Lagrangians in their commutative and noncommutative regimes. The corresponding noncommutative path integral is presented. This method is illustrated by two quantum-mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.