Broiling deBroglie ... Phun Physics for Phreaks & Geeks

Merely to shake up our audience, because that is, after all, why we are here,  I offer something for the physicists, engineers and other geek savants to ruminate about.  I thought I would just copy some personal notes out of a Word doc I typed up a while ago, and voila, here it would appear to enlighten and enchant you --  but no, it was vastly more painful than that. Let me tell you,  you haven't lived (or died) till you've tried to put equations into Blogger... this was a learning experience, and there is nothing intuitive about the formatting you are seeing here. So appreciate it.  Or else.

Let’s say you have two electrons in free space, and one of them is in motion towards the other electron. At rest, each has a relativistic mass-energy,

                                  (1)

Now assume each electron has a corresponding electromagnetic frequency f_e associated with a single photon,

                                       (2)

                                   (3)


Where h is Planck’s constant. For an electron, this frequency is quite high – about 10²⁰ Hz. 100 million terahertz.

Assume a photon of energy E_e can be bound in a circular loop in some way to comprise the electron. (Think of annihilating an electron and positron into two gamma rays—what were the gammas before the annihilation?) By analogy, think of the binding as something like total internal reflection within a dielectric sphere, though my idea is more like a spatial distortion caused by the intense localized concentration of electromagnetic energy.

Further, assume the electrostatic Coulomb field of the electron is modulated at the frequency f_e, and think of the static field as radiating outward as a spherical evanescent wave (a near-field effect) —that is, without releasing any electromagnetic energy. This is by analogy to the external evanescent field around an optical glass fiber. This isn’t entirely unreasonable (it can even explain electron spin and magnetic moment), but a more detailed explanation requires adapting a field solution for a fiber optic waveguide to spherical coordinates of a dielectric sphere, taking the limit for a sphere radius of zero. (Only in this limit does the evanescent field of a sphere exhibit the 1/r² static field dependences of a Coulomb force.)

If you’re really interested in evanescent fields, you can look at http://en.wikipedia.org/wiki/Evanescent_wave, though the treatment is poor. A better reference is a good graduate textbook on electromagnetic fields (eg, Ramo or Jackson). For now, simply assume the Coulomb field can be derived as the evanescent field of a bound photon in a dielectric sphere of zero radius.

Coulomb’s Law for the force of interaction between the static fields of two electrons is

               (4)


If the static fields of the electrons are modulated at frequency f_e, it seems reasonable to say that the Coulomb force is modulated as well. We can set aside for the moment exactly how it is modulated (it might be discrete, for instance, rather than sinusoidal). The main goal is to see if there is a wavelength for the force coupling, on the premises given.

If you grant the premises, at slow speed the electron moving at speed v “sees” the radiating wavefronts of the Coulomb field from the static electron as dopplered up to the frequency

                             (5)


The premise is the Coulomb force experienced by the moving electron is envelope modulated at the frequency difference Df   (greek delta evaded my formatting skills) between the oncoming wavefronts at frequency f^'_e, and its own internal frequency f_e. This is our "matter wave”.

                    (6)


By analogy to an electronic mixer used to subtract frequencies in a radio frequency (RF) heterodyning stage, this is the beat frequency between the two electrons. If you could observe the interacting wavefronts of the (hypothetical) modulated static fields of the two electrons, you would see an envelope modulation corresponding to this beat frequency, and the envelope would be traveling at the speed of light c with a wavelength

(7)

                                                (8)


Which is, voila, the deBroglie formula (click that link whether you think you know how to pronounce "deBroglie" or not), where p is the non-relativistic momentum of the moving electron. It is worth noting that the same result would be obtained for however many photons we assume to comprise the electron rest energy. It is only important that electrons are the same. The same result would apply for protons to protons. Or any other particle to another particle like itself.

There are a number of problems with my assumptions, but it is intriguing that they give deBroglie’s matter wavelength from such a simple derivation, and worth speculating whether the idea has more substance.

In my next post for this exciting thrill-ride of a thread, I'll show you how to derive all the basic relations of special relativity in 2 pages.   From first principles.  Really.   But before that I shall probably meet the same fate as Hypatia in Agora.