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 Mathematical Physics

Feynman path integrals and quantum mechanics as h/→0

Aubrey Truman

Published Oct 1, 1976• 61 citations
Publisher

Abstract

When the space of paths is a certain Hilbert space H, we show how to extend the Feynman path integral ℱ of DeWitt and Albeverio and Hoegh-Krohn. Our extension enables us to integrate a wider class of functionals on H. We establish a new representation for the wavefunction in nonrelativistic quantum mechanics—the quasiclassical representation. Using our extension of ℱ and the quasiclassical representation, we discuss the problem of obtaining classical mechanics as the limiting case of quantum mechanics when h/→0.

DOI
10.1063/1.522806

Related papers

New path integral representation of the quantum mechanical propagator

M. Roncadelli

A new path integral representation of the quantum mechanical propagator is proposed, which is even more intimately connected with classical mechanics than the Feynman path integral. The emergence of the semiclassical propagator is discussed in detail. It is argued that those paths which actually contribute in the new path integral have a very simple relation with the classical dynamical trajectories in configuration space.

Journal of Physics A Mathematical and General•Aug 21, 1992•6 citations

A New Form of Path Integral for the Coherent States Representation and its Semiclassical Limit

Luis C. dos Santos, Marcus A. M. de Aguiar

The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in \cite{Bar01} for the two corresponding path integral forms suggested by Klauder and Skagerstan in \cite{Klau85}. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.

arXiv:quant-ph/0412080•Dec 10, 2004•0 citations

A new form of path integral for the coherent states representation and its semiclassical limit

Luana Carolina Santos, Marcus A. M. de Aguiar

The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in [1] for the two corresponding path integral forms suggested by Klauder and Skagerstan in [2]. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.

Brazilian Journal of Physics•Mar 1, 2005•4 citations

The Classical Limit of Quantum Mechanics

Velton Wade Hefley

The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.

•Dec 1, 1977•1 citation

Feynman maps without improper integrals

Pavel Exner, G. I. Kolerov

The Feynman maps introduced first by Truman are examined. The domain considered here consists of the Fresnel-integrable functions in the sense of Albeverio and Hoegh-Krohn; extensions to wider classes of functions will be studied in the following paper. The original definition of the F-maps is slightly modified: we start from the underlying measures on the Hilbert space of paths in order to avoid use of improper integrals. Some new properties of the F-maps are derived. In particular, the dominated convergence theorem is shown to be not valid for the F1-map (or Feynman integral); this fact is of a certain importance for the classical limit of quantum mechanics.

•Jan 1, 1980•0 citations

Introduction of Feynman's path integral theory into engineering physics

Zhou Pei

Expressing quantum mechanics with Feynman's path integral theory, the classical mechanics is bringed into the theoretical frame of quantum mechanics as a limiting case of S.

College Physics•Jan 1, 2000•0 citations
When the space of paths is a certain Hilbert space H, we show how to extend the Feynman path integral ℱ of DeWitt and Albeverio and Hoegh-Krohn. Our extension enables us to integrate a wider class of functionals on H. We establish a new representation for the wavefunction in nonrelativistic quantum mechanics—the quasiclassical representation. Using our extension of ℱ and the quasiclassical representation, we discuss the problem of obtaining classical mechanics as the limiting case of quantum mechanics when h/→0.
The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in \cite{Bar01} for the two corresponding path integral forms suggested by Klauder and Skagerstan in \cite{Klau85}. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.
The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in [1] for the two corresponding path integral forms suggested by Klauder and Skagerstan in [2]. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.
The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.
The Feynman maps introduced first by Truman are examined. The domain considered here consists of the Fresnel-integrable functions in the sense of Albeverio and Hoegh-Krohn; extensions to wider classes of functions will be studied in the following paper. The original definition of the F-maps is slightly modified: we start from the underlying measures on the Hilbert space of paths in order to avoid use of improper integrals. Some new properties of the F-maps are derived. In particular, the dominated convergence theorem is shown to be not valid for the F1-map (or Feynman integral); this fact is of a certain importance for the classical limit of quantum mechanics.