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

On the semi-classical expansion in quantum mechanics for arbitrary Hamiltonians

Maurice M. Mizrahi

Published May 1, 1976• 37 citations
OpenAlex

Abstract

The purpose of this paper is to extend to a very wide class of Hamiltonians the range of validity of a well-known and oft-used expression for the semi-classical (WKB) approximation to the quantum-mechanical propagator K((q sub b),(t sub b);(q sub a),(t sub a)) identically equal to ((q sub a),(t sub b)/(q sub a),(t sub a)), or probability amplitude that a particle at (q sub a) at time (t sub a) will arrive at (q sub b) at time (t sub b).

Related papers

On the WKB approximation to the propagator for arbitrary Hamiltonians

Maurice M. Mizrahi

This paper presents a general expression for the WKB approximation to the propagator corresponding to an arbitrary Hamiltonian operator H. For example, if the correspondence rule used to pass from the classical Hamiltonian Hc to H is such that it associates aPiQ j +(1−a)Q jPi to piq j, then the formula gives KWKB=KVVexp {( 1/2 −a)ℱT(∂2Hc/∂qi∂pi )(qc(t),pc(t),t) dt}, where KVV≡(2πih/)−n/2 (det M)1/2exp(iSc/h/) is Van Vleck’s well-known formula, Sc being the action functional evaluated at the classical path (qc,pc) and Mij ≡−∂2Sc/∂qa i∂qbj. More generally, the formula presented here applies to any system with n degrees of freedom described by a function f(x,t) whose time evolution is given by (H(x,k∂/∂x,t)+k∂/∂t) f(x,t)=0, regardless of the form of H. The Schrödinger equation of quantum mechanics and the Fokker–Planck equation of diffusion are obvious examples. Many examples are discussed. This generalizes results obtained in a previous publication [J. Math. Phys. 18, 786–90 (1977)].

Journal of Mathematical Physics•Jan 1, 1981•14 citations

The General Expansion of WKB Approximation method to the Propagator for Arbitrary Hamiltonians

Zhiqiang Hu

The general method of path integrals in quantum mechanics introduced by R.P.Feynman is applied in this paper to discuss the amplitude of transition probability in quantum mechanics, that is, the subject of the propagator.Mauriel M.Mizrahi's semiclassical (WKB) approximation method has been used in studying the general expansion in detail for the propagator corresponding to an arbitrary Hamiltonian operator H. ?? where S_c is the action functional evaluated at the classical path (q_c, p_c) , ρ is a constant, ?? Maurice M.Mizrahi's solution of the K_(WKB) expansion has been generalized, and another linear independent particular solution is obtained.This is the solution when 0~2 = det M equals zero.The K_(WKB) can be applied to all systems corresponding to an arbitrary Hamiltonian operator.

Huadong Shifan Daxue xuebao. Ziran kexue ban•Jan 1, 1983•0 citations

Path integrals and the WKB approximation

Barry R. Holstein, Arthur R. Swift

The WKB approximation is demonstrated to be formally identical to the semiclassical limit of the Feynman path integral propagator. In the limit that the path integral propagator is dominated by the classical particle trajectory, it is identical to the propagator calculated directly from WKB wave functions in the stationary phase approximation.

American Journal of Physics•Sep 1, 1982•35 citations

The WKB approximation for a linear potential and ceiling

Todd Zapata

The physical problem this thesis deals with is a quantum system with linear\npotential driving a particle away from a ceiling (impenetrable barrier). This thesis\nwill construct the WKB approximation of the quantum mechanical propagator. The\napplication of the approximation will be for propagators corresponding to both initial\nmomentum data and initial position data.\nAlthough the analytic solution for the propagator exists, it is an indefinite integral of Airy functions and di±cult to use in obtaining probability densities by numerical integration or other schemes considered by the author. The WKB construction\nis less problematic because it is representable in exact form, and integration schemes\n(both numerical and analytic) to obtain probability densities are straightforward to\nimplement. Another purpose of this thesis is to be a starting point for the construction of WKB propagators with general potentials but the same type of boundary,\nimpenetrable barrier.\nResearch pertaining to this thesis includes determining all classical paths and\nconstraints for the one-dimensional linear potential with ceiling, and using these\nequations to construct the classical action, and hence the WKB approximation. Also,\nevaluation of final quantum wave functions using numerical integration to check and\nbetter understand the approximation is part of the research.\nThe results indicate that the validity of the WKB approximation depends on the\ntype of classical paths (i.e. the initial data of the path) used in the construction. Specifically, the presence of the ceiling may cause the semi-classical wave packets\nto become vanishingly small in one representation of initial classical data, while not\neffecting the packets in another. The conclusion of this phenomenon is that the\nrepresentation where the packets are not annihilated is the correct representation.

•May 15, 2009•3 citations

On the semi-classical expression of the quantum-mechanical propagator

Paolo Moretti, G. Bartalini

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B•Feb 1, 1992•0 citations

Classical Limit and the WKB Approximation

Lowell S. Brown

We prove that a wave-packet solution in the WKB approximation to the Schrödinger equation follows the classical trajectory, and we discuss the nature of this solution.

American Journal of Physics•Mar 1, 1972•13 citations
The purpose of this paper is to extend to a very wide class of Hamiltonians the range of validity of a well-known and oft-used expression for the semi-classical (WKB) approximation to the quantum-mechanical propagator K((q sub b),(t sub b);(q sub a),(t sub a)) identically equal to ((q sub a),(t sub b)/(q sub a),(t sub a)), or probability amplitude that a particle at (q sub a) at time (t sub a) will arrive at (q sub b) at time (t sub b).
This paper presents a general expression for the WKB approximation to the propagator corresponding to an arbitrary Hamiltonian operator H. For example, if the correspondence rule used to pass from the classical Hamiltonian Hc to H is such that it associates aPiQ j +(1−a)Q jPi to piq j, then the formula gives KWKB=KVVexp {( 1/2 −a)ℱT(∂2Hc/∂qi∂pi )(qc(t),pc(t),t) dt}, where KVV≡(2πih/)−n/2 (det M)1/2exp(iSc/h/) is Van Vleck’s well-known formula, Sc being the action functional evaluated at the classical path (qc,pc) and Mij ≡−∂2Sc/∂qa i∂qbj. More generally, the formula presented here applies to any system with n degrees of freedom described by a function f(x,t) whose time evolution is given by (H(x,k∂/∂x,t)+k∂/∂t) f(x,t)=0, regardless of the form of H. The Schrödinger equation of quantum mechanics and the Fokker–Planck equation of diffusion are obvious examples. Many examples are discussed. This generalizes results obtained in a previous publication [J. Math. Phys. 18, 786–90 (1977)].
The general method of path integrals in quantum mechanics introduced by R.P.Feynman is applied in this paper to discuss the amplitude of transition probability in quantum mechanics, that is, the subject of the propagator.Mauriel M.Mizrahi's semiclassical (WKB) approximation method has been used in studying the general expansion in detail for the propagator corresponding to an arbitrary Hamiltonian operator H. ?? where S_c is the action functional evaluated at the classical path (q_c, p_c) , ρ is a constant, ?? Maurice M.Mizrahi's solution of the K_(WKB) expansion has been generalized, and another linear independent particular solution is obtained.This is the solution when 0~2 = det M equals zero.The K_(WKB) can be applied to all systems corresponding to an arbitrary Hamiltonian operator.
The physical problem this thesis deals with is a quantum system with linear\npotential driving a particle away from a ceiling (impenetrable barrier). This thesis\nwill construct the WKB approximation of the quantum mechanical propagator. The\napplication of the approximation will be for propagators corresponding to both initial\nmomentum data and initial position data.\nAlthough the analytic solution for the propagator exists, it is an indefinite integral of Airy functions and di±cult to use in obtaining probability densities by numerical integration or other schemes considered by the author. The WKB construction\nis less problematic because it is representable in exact form, and integration schemes\n(both numerical and analytic) to obtain probability densities are straightforward to\nimplement. Another purpose of this thesis is to be a starting point for the construction of WKB propagators with general potentials but the same type of boundary,\nimpenetrable barrier.\nResearch pertaining to this thesis includes determining all classical paths and\nconstraints for the one-dimensional linear potential with ceiling, and using these\nequations to construct the classical action, and hence the WKB approximation. Also,\nevaluation of final quantum wave functions using numerical integration to check and\nbetter understand the approximation is part of the research.\nThe results indicate that the validity of the WKB approximation depends on the\ntype of classical paths (i.e. the initial data of the path) used in the construction. Specifically, the presence of the ceiling may cause the semi-classical wave packets\nto become vanishingly small in one representation of initial classical data, while not\neffecting the packets in another. The conclusion of this phenomenon is that the\nrepresentation where the packets are not annihilated is the correct representation.