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PATH INTEGRALS IN POLAR COORDINATES: CHARGED PARTICLES IN A UNIFORM MAGNETIC FIELD.

Akira Inomata

Published Feb 1, 1967• 0 citations
OpenAlex

Abstract

The statistical density matrix for a system of charged particles in a uniform magnetic field is derived by Feyman's path integral method. The integrations over all paths are directly carried out by use of cylindrical coordinates. As limiting cases, the density matrices for free particles, rigid rotators, and harmonic oscillators are also discussed. (Author)

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arXiv:1908.05226•Aug 13, 2019•1 citation
The statistical density matrix for a system of charged particles in a uniform magnetic field is derived by Feyman's path integral method. The integrations over all paths are directly carried out by use of cylindrical coordinates. As limiting cases, the density matrices for free particles, rigid rotators, and harmonic oscillators are also discussed. (Author)
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.
In the present paper the author evaluates the path integral of a charged anisotropic Harmonic Oscillator (HO) in crossed electric and magnetic fields by two alternative methods. Both methods enable a rather formal calculation and circumvent some mathematical delicate issues such as the occurrence of an infinite Normalization constant and ambiguities with path integral calculations when magnetic fields are present. The \emph{1st1^{\text{st}}1st} method uses complex Fourier series and a regularization scheme via the Riemann-ζζζ-function. The \emph{2nd2^{\text{nd}}2nd} method evaluates the path integral by transforming the Lagrangian to a uniformly rotating system. The latter method uses the fact that the Lorentz- and Coriolis force have the same functional form. Both forces cancel each other within the rotating system given that it rotates with Larmor frequency ωLω_{L}ωL​. This fact simplifies considerably the calculation of the path integral.