The representation of radiation reaction in wave mechanics

Abstract

It is well known that the wave mechanical ψ equation leads to the conclusion that the centroid of the wave mechanical electron should move according to the classical electrodynamic equation of motion in which, however, the terms representing what is commonly called radiation reaction are absent. If v is the velocity of the electron, the classical rate of change of momentum is md dt { v ( I − v 2 c 2 ) 1 2 } . The equation of motion including radiation reaction terms may be regarded as obtainable by replacing this quantity by one obtained by operating upon it with the operator P −1 P={I− α 1kd dt + α 2d dt ( kd dt )−·} − where α 1, α 2, etc., are constants and k = ( I − v 2 c 2 ) −1 2 . The main purpose of the paper is to show that if there be any relativistically invariant ψ equation which leads to the classical equation of motion without radiation reaction terms, then by replacing the vector and scalar potentials U and ϕ in that equation by P( U) and P( ϕ), a relativistically invariant equation of motion will be obtained including the radiation reaction terms, provided that the d dt in P be now regarded as ∂ ∂t + u · grad, where u is the velocity of the wave mechanical density distribution at a point. The purpose is to use the power to produce the equation of motion as a criterion for suggesting the proper modification of the ψ equation to apply in those cases where, on the classical theory, the electron would suffer great acceleration, as in ionization by rapidly moving corpuscles.