15. The Photon

In Sections 2-14 it was shown how Newtonian mechanics could be based in electromagnetic theory. The theory was also successfully tested at relativistic velocities. We will now go to a new subject and, in fact, a new particle. When Maxwell completed his famous equations, one of the most intriguing and surprising results to come out of them was that the speed of light could be derived from two constants that were independently measured within electromagnetism itself. That is c=1/(em)½ where e is the permitivity used for Coulomb's law and m is the permeability used for the Biot-Savart law. This seemed to indicate that light was propagated in the same medium as was radio phenomenon. However, with the advent of photon theory, where light was a particle, this correspondence was essentially cancelled.

We will show in this section that light really is a wave, but a wave where a single pulse can propagate without spreading. This is shown quite simply in the first derivation. The next derivation gives a somewhat ideal example of a photon-like soliton/particle.

Consider Eq.(3-2) again. If Ez=0, Then the equation becomes

(1)


This propagation is in one direction. In other words, this type of cylindrical field propagates without spreading. If a single pulse was created, it would move along like a particle at the speed of light. We can now define three types of propagation in the space. Typical classical radiation of the electric field; a particle-like propagation which depending on its symmetry can travel any velocity less than the speed of light; A photon-like radiation (provided it is emitted in a single pulse) where a non attenuating wave with cylindrical symmetry propagates at the speed of light1. There is still a large degree of freedom in the actual form of the field. The cylindrical field must decrease radially as 1/R2 so that the energy will be finite. It must have a region of negative field and a region of positive field due to the way the photon is created (see Section 23). In consideration of these limitations, we can write

(2)


where k and h are constants to be determined and the value over z covers only one wavelength. This is then a pulse with the maximum frequency contribution being c/l . Note that at z=0 and z=l there are discontinuities, but we will assume that there are variations at these points that satisfy continuity, but are negligible in relation to the rest of the field. The energy of the field can be determined by

(3)


This yields

(4)


We want to compare this with the energy in the electric field2 of a photon given by:

(5)


Note that the energy is strictly related to the frequency as opposed to the amplitude of the field. However, if we let h=l, then this pulse will have the same frequency relation. In other words, we are requiring that the spread of the radial field always be proportional to the central wavelength (largest Fourier transform term) of the pulse. Then k is a constant defined by

(6)


Thus Eq.(2) with this value of k defines the field of a photon like propagation in the space at dimensions comparable to an actual photon. This cylindrical symmetric pulse will propagate without attenuation at the speed of light. It should be noted here that the photon is a peculiar object. It is not a particle or a wave. In the direction parallel to its propagation there is zero divergence. It thus acts as a wave in that dimension. Perpendicular to its motion, it has a divergence and thus acts as a particle. Though the electric field is reversed over the single wavelength of the photon, it does not represent a dipole since there is no real charge present. Laterally, the field does not fall off as a dipole field but as a monopole field. Longitudinally, the field is of finite extent.
1. Unlike the soliton particle, where we had to postulate a mechanism to transport the longitudinal field, the photon has no longitudinal field and thus the initial postulates fully support the process. No field in z develops so the equation remains separable.

2. This assumes that the total energy is x=hf where h is Plank's constant (see for example Griffiths1 481). The division of energy between the electric and magnetic field in a photon is discussed in the next section.
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