EVALUATION OF THE FEYNMAN S PROPAGATOR BY MEANS OF THE QUANTUM HAMILTON -JACOBI EQUATION Mario Fusco Girard

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EVALUATION OF THE FEYNMAN’S PROPAGATOR BY MEANS OF

THE QUANTUM HAMILTON-JACOBI EQUATION

Mario Fusco Girard

Department of Physics “E. R. Caianiello”

University of Salerno (Italy)

Electronic address: mfuscogirard@gmail.com

ABSTRACT

It is shown that the complex phase of the Feynman’s propagator is a solution of the quantum

Hamilton-Jacobi equation, i.e. it is the quantum Hamilton’s principal function (or quantum action).

The propagator so can be computed either by means of the path integration, or by the way of that

equation. This is analogous to what happens in classical mechanics, where the Hamilton’s principal

function can be computed or by integrating the lagrangian along the extremal paths, or as solution

of a partial differential equation, namely the classical Hamilton-Jacobi equation. If the path is

decomposed in the classical one and quantum fluctuations, the contribution of these latter satisfies

a non-linear partial differential equation, whose coefficients depend on the classical action. When

the contribution of the quantum fluctuations depend only on the time, it can be computed by means

of a simple integration. The final results for the propagators in this case are equal to the Van Vleck-

Pauli-Morette expressions, even though the two derivations are quite different.

1. THE METHOD

As well known, the propagator is the fundamental quantity in the Feynman’s space-time approach

to non-relativistic quantum mechanics [1, 2]. It gives the quantum amplitude for a particle to go

from a point xA at time tA, to xB at time tB, as a path integral, i.e. a sum of contributions φ[x(t)] from

each path connecting the two points in the time tB – tA (for simplicity we adopt a one-dimensional

notation):

(1),

here L(x(t), v(t), t) is the classical Lagrangian.

The propagator, originally named the kernel by Feynman, is the Green function of the Schrödinger’s

equation [2 - 6] :

(2),

where H is the Hamiltonian operator.

When xB≠ xA and tB> tA, the propagator therefore satisfies the time-dependent Schrödinger’s

equation, and in this way it can be connected to the Quantum Hamilton-Jacobi Equation (QHJE) [7,

8]. This latter, fully equivalent to the Schrödinger’s equation, is the starting point of the WKB

approximation, and appears when one tries to find solutions in exponential form

(3)

of the Schrödinger’s equation for a particle of mass m in a potential V(x):

(4).

S(x, t) in (3) is a complex function and C is a constant.

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摘要：

EVALUATIONOFTHEFEYNMAN’SPROPAGATORBYMEANSOFTHEQUANTUMHAMILTON-JACOBIEQUATIONMarioFuscoGirardDepartmentofPhysics“E.R.Caianiello”UniversityofSalerno(Italy)Electronicaddress:mfuscogirard@gmail.comABSTRACTItisshownthatthecomplexphaseoftheFeynman’spropagatorisasolutionofthequantumHamilton-Jacobiequation,...

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EVALUATION OF THE FEYNMAN S PROPAGATOR BY MEANS OF THE QUANTUM HAMILTON -JACOBI EQUATION Mario Fusco Girard

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