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理论物理通讯:英文版

Alternative Derivation of the Propagator in Polar Coordinates by Feynman's Physical Interpretation of the Characteristic Function

Qian Qian, Shang-Wu, Чан, King-Man, Gu, Zhiyu

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

This article revisits Feynman's characteristic function, and points out the insight and usefulness of hisphysical interpretation. As an example, the tedious and rather long derivation of the propagator in polar coordinatescan be easily and clearly obtained by merely using Feynman's physical intepretation of the characteristic function andsome well-known results of central force problem.

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This article revisits Feynman's characteristic function, and points out the insight and usefulness of hisphysical interpretation. As an example, the tedious and rather long derivation of the propagator in polar coordinatescan be easily and clearly obtained by merely using Feynman's physical intepretation of the characteristic function andsome well-known results of central force problem.
In this paper, we present a mathematical derivation of the Schrödinger equation departing from only two axioms. We also show that, using this formal derivation process, it is possible to directly derive the Schrödinger equation in generalized curvilinear coordinate systems. This derivation is also shown to be equivalent to Feynman’s path integral approach, but goes further, allowing us to mathematically derive the Bohr-Sommerfeld quantization rules. The use of a small parameter, both in the present derivation, where it is δr, and Feynman’s derivation, where it is ϵ = δt, is also clarified in terms of the Central Limit Theorem. Therefore, the article makes a didactic transposition of the topic of quantization, allowing it to be addressed in the context of teaching Quantum Mechanics. The epistemological importance of axiomatic approaches for the mathematical derivation and the interpretation of the symbols of the theory is also considered.
Based on the equivalence between momentum representation and coordinate representation,an alternative method is presented,where the momentum representation and the Feynman's path integral are employed to calculate the propagator function of a particle in a constant force field,then the result is transformed into the coordinate representation. We present a way to solve the Feynman's path integral,which exhibits the high value of momentum representation if the solution of one- dimensional motion of a particle in a constant force field.
Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central-force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse-square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.
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.