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Journal of Mathematical Physics

Summation over Feynman Histories in Polar Coordinates

David Peak, Akira Inomata

Published Aug 1, 1969• 225 citations
Publisher

Abstract

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.

DOI
10.1063/1.1664984

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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.
Abstract A central coordinate system is investigated within the framework of the Wigner-Dunkl deformation of quantum mechanics, treated using Feynman’s path integral approach. This approach involves performing path integration of a Lagrangian in polar coordinates, with an effective potential term that depends on both radial and angular coordinates. The success in separating variables within the two-dimensional Dunkl system facilitated obtaining exact expressions for the propagators associated with the free particle, inverse square potential, harmonic oscillator, and Coulomb potential. The energy eigenvalues and their corresponding wave functions are also determined.
We consider central-potential scattering and determine a path-integral representation for the SSS matrix in polar coordinates. This is obtained by transforming to polar coordinates a Cartesian form of the nonrelativistic SSS matrix given by Campbell et al., and implementing an idea of Faddeev to obtain the appropriate asymptotic conditions. Our results are applied to scattering in an inverse-square potential to determine the correct phase shifts as well as the SSS matrix.