1. Introduction: the Spinning Particle Model
1. Introduction: the Spinning Particle Model
In the previous article (Spinning Particle Model (I): a classical model of the electron spin, called Part I in this article) we have derived (both numerically and graphically) the Spinning Particle Model, describing the dynamics of the electron as a classical Dirac particle with spin (with angular momentum) providing the equations and code to model the electron. This model satisfies the Dirac equation when quantized.
In the second article (Spinning Particle Model (II): Electron with spin in ElectroMagnetic fields), we have focused on modelling the interaction of the electron with electric and magnetic fields, deriving the expressions of the interaction and solving and visualizing them with the code.
In this third article of the series we will derive and model the interaction between two electrons and also in the presence of electromagnetic fields.
We will refer to the formula numbers (XX) in the previous articles as Part I.(XX)
In the second article (Spinning Particle Model (II): Electron with spin in ElectroMagnetic fields), we have focused on modelling the interaction of the electron with electric and magnetic fields, deriving the expressions of the interaction and solving and visualizing them with the code.
In this third article of the series we will derive and model the interaction between two electrons and also in the presence of electromagnetic fields.
We will refer to the formula numbers (XX) in the previous articles as Part I.(XX)
As the previous article, this article is based on the work by Martin Rivas and most of the text and figures are extracts from his papers, detailed references are listed in Section 12.
The Spinning Particle Model has been derived through a general formalism, without any assumption about the shape or behaviour of the electron, from the three fundamental principles used: Restricted Relativity principle, Variational principle and Atomic principle.
We have already seen that the key feature of the Spinning Particle Model is that it simplifies the particle model to tracking just one point, whose motion determines all observable properties:
We only need to describe the evolution of the point,, the Center of Charge (CC), where the external forces are defined, and all observables: energy, linear momentum, angular momentum, Center of Mass (CM) position, angular velocity and spins are expressed in terms of this point and its time derivatives. The Center of Mass (CM, ) is a different point than the Center of Charge (CC, ), and is a function of the CC position. The Center of Mass satisfies Newton type equations in terms of the external forces evaluated at the position of the CC.
We have already seen that the key feature of the Spinning Particle Model is that it simplifies the particle model to tracking just one point, whose motion determines all observable properties:
We only need to describe the evolution of the point,
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The Center of Charge is moving at the speed of light (| | = c) and the Center of Mass is moving at a speed || < c. In the free motion the CC describes a circle of radius =, contained in a plane orthogonal to the spin. The CC is continuously spinning around the CM, describing a circle when ||=0 (CM observer) and an helix for the laboratory observer when 0<||<c.The complete dynamical equations for the evolution of the Center of Mass and Center of Charge can be expressed in terms of dimensionless classical variables.
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