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

arXiv:0711.3544 · Symmetry Integrability and Geometry Methods and Applications

Alternative Method for Determining the Feynman Propagator of a Non-Relativistic Quantum Mechanical Problem

M. Moshińsky

Published Nov 22, 2007• 9 citations
arXivHTMLPDF

Abstract

A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman.The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later.In our approach, with the help of a Laplace transform, a direct way to get the energy dependent Green function is presented, and the propagator can be obtained later with an inverse Laplace transform.The method is illustrated through simple one dimensional examples and for time independent potentials, though it can be generalized to the derivation of more complicated propagators.

DOI
10.3842/sigma.2007.110

Related papers

Alternative method for determining the Feynman propagator of a relativistic quantum mechanical problem

M. Moshińsky, Emerson Sadurní

The authors, together with A. del Campo, developed an alternative method for determining the Feynman propagator [1] for a non-relativistic problem. One started with the time dependent Schroedinger equation for the problem. Carried out a Laplace transform with respect to time to get the equation for the energy dependent Green function and derived it explicitly. We then carried out the inverse Laplace transform in the energy to get Feynman propagator. In this paper we carry out the same programme for a relativistic problem associated with the one dimensional Dirac equation of a free particle and the Dirac oscillator proposed by Moshinsky and Szczepaniak [2] twenty years ago.

Revista Mexicana de Física•Dec 1, 2008•4 citations

A systematic approach for obtaining the Green functions of time-dependent Schrödinger equations by Fourier transform

Gin-yih Tsaur, Jyhpyng Wang

Abstract The Green function method is a powerful technique for solving the initial value problem in quantum mechanics. Once the Green function is solved the whole wavefunction evolution is represented in a concise form and can be evaluated conveniently by numerical integration. We present a method for constructing the Green functions systematically which is different from the conventional methods of eigenfunction expansion or path integration. By using variable changing, function substitution, and Fourier transforms, the time dependent Schrödinger equations can be simplified and the solutions for the simplified equations can be easily derived. We then obtain the Green functions for the original equations by the reverse transforms. The method is demonstrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator, where the Green function for the centripetal barrier oscillator has not been solved previously by conventional methods. The method and examples illustrated in this paper can be utilised to strengthen undergraduate courses on quantum mechanics and/or partial differential equation.

European Journal of Physics•May 10, 2016•1 citation

Operator method for finding new propagators from old

Arlen Anderson

A new approach to finding the exact Green's function to the time-dependent Schr\"odinger equation is described. The approach is conceptually simple: one seeks an operator which when commuted through the Schr\"odinger operator transforms it into the Schr\"odinger operator of a soluble problem. This approach provides an alternative to both the eigenfunction expansion of the propagator and the path integral. The first example of an exact propagator represented as a ``sum over classical paths'' which is not equal to the Gaussian approximation of the path integral is found.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jan 15, 1988•18 citations

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

Propagator for the δ-function potential moving with constant velocity

Bin Kang Cheng, M. G. E. da Luz

We first evaluate the exact propagator for a \ensuremath{\delta}-function potential moving with constant velocity by summing over the spectrum of eigenstates and by calculating the path integral directly. The ``one-time Green's functions'' are then derived from the Fourier transform of the propagator obtained. We finally investigate the propagator through an asymptotic approximation.

Physical Review A•Jun 1, 1993•10 citations

A sum-over-paths approach to one-dimensional time-independent quantum systems

Massimiliano Malgieri, Pasquale Onorato, Anna De Ambrosis

We present an alternative treatment for simple time-independent quantum systems in one dimension, which can be used in the context of an elementary introduction to quantum physics using the Feynman approach. The method is based on representation of the energy-dependent propagator (or Green function) as a sum of complex amplitudes over all possible paths, classical and non-classical, at fixed energy. We treat both confined and open systems with piecewise-constant potentials, obtaining exact results. We introduce an approximation scheme to extend the method to smooth potentials, recovering the Van Vleck-Gutzwiller propagator. Finally, we discuss the educational application of the method.

American Journal of Physics•Aug 20, 2016•6 citations
A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman.The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later.In our approach, with the help of a Laplace transform, a direct way to get the energy dependent Green function is presented, and the propagator can be obtained later with an inverse Laplace transform.The method is illustrated through simple one dimensional examples and for time independent potentials, though it can be generalized to the derivation of more complicated propagators.
The authors, together with A. del Campo, developed an alternative method for determining the Feynman propagator [1] for a non-relativistic problem. One started with the time dependent Schroedinger equation for the problem. Carried out a Laplace transform with respect to time to get the equation for the energy dependent Green function and derived it explicitly. We then carried out the inverse Laplace transform in the energy to get Feynman propagator. In this paper we carry out the same programme for a relativistic problem associated with the one dimensional Dirac equation of a free particle and the Dirac oscillator proposed by Moshinsky and Szczepaniak [2] twenty years ago.
Abstract The Green function method is a powerful technique for solving the initial value problem in quantum mechanics. Once the Green function is solved the whole wavefunction evolution is represented in a concise form and can be evaluated conveniently by numerical integration. We present a method for constructing the Green functions systematically which is different from the conventional methods of eigenfunction expansion or path integration. By using variable changing, function substitution, and Fourier transforms, the time dependent Schrödinger equations can be simplified and the solutions for the simplified equations can be easily derived. We then obtain the Green functions for the original equations by the reverse transforms. The method is demonstrated for the linear potential, the harmonic oscillator, the centrifugal potential, and the centripetal barrier oscillator, where the Green function for the centripetal barrier oscillator has not been solved previously by conventional methods. The method and examples illustrated in this paper can be utilised to strengthen undergraduate courses on quantum mechanics and/or partial differential equation.
A new approach to finding the exact Green's function to the time-dependent Schr\"odinger equation is described. The approach is conceptually simple: one seeks an operator which when commuted through the Schr\"odinger operator transforms it into the Schr\"odinger operator of a soluble problem. This approach provides an alternative to both the eigenfunction expansion of the propagator and the path integral. The first example of an exact propagator represented as a ``sum over classical paths'' which is not equal to the Gaussian approximation of the path integral is found.
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.
We present an alternative treatment for simple time-independent quantum systems in one dimension, which can be used in the context of an elementary introduction to quantum physics using the Feynman approach. The method is based on representation of the energy-dependent propagator (or Green function) as a sum of complex amplitudes over all possible paths, classical and non-classical, at fixed energy. We treat both confined and open systems with piecewise-constant potentials, obtaining exact results. We introduce an approximation scheme to extend the method to smooth potentials, recovering the Van Vleck-Gutzwiller propagator. Finally, we discuss the educational application of the method.