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Hamilton–Jacobi approach to photon wave mechanics: near‐field aspects
Author(s) -
KELLER O.
Publication year - 2008
Publication title -
journal of microscopy
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.569
H-Index - 111
eISSN - 1365-2818
pISSN - 0022-2720
DOI - 10.1111/j.1365-2818.2008.01909.x
Subject(s) - wave mechanics , photon , hamilton–jacobi equation , physics , classical mechanics , field (mathematics) , quantum mechanics , mathematics , pure mathematics
Summary After having briefly reviewed the Hamilton–Jacobi theory of classical point‐particle mechanics, its extension to the quantum regime and the formal identity between the Hamilton–Jacobi equation for Hamilton's characteristic function and the eikonal equation of geometrical optics, an eikonal theory for free photons is established. The space‐time dynamics of the photon is described on the basis of the six‐component Riemann–Silberstein energy wave function. Form‐identical eikonal equations are obtained for the positive and negative helicity dynamics. Microscopic response theory is used to describe the linear photon–matter interaction. In the presence of matter the free‐photon concept is replaced by a quasi‐photon concept, and there is a quasi‐photon for each of the two helicity states. After having established integro–differential equations for the wave functions of the two quasi‐photons, the eikonal conditions for the quasi‐photons are determined. It appears that the eikonal condition contains complicated space integrals of the gradient of the eikonal over volumes of near‐field domain size. In these space integrals the dynamics of the electrons (matter particles) appears via transverse transition current densities between pairs of many‐body states. Generalized microscopic polarization and magnetization fields are introduced to establish the connection between the quasi‐photon and macroscopic eikonal theories.