Date sön 20 maj 2018

If you read about Mill's Brilliant Light power in the wikipedia article you will notice that it is associated with pseudoscience. And you will get the impression that all are wrong. But is it. We can take one fact that is the base of whole theory, namely that he constructs a set of orbitals that has non radiative properties and hence has all needed mathematical properties to explain the discrete orbitals we today understand the atoms have. And this with only special solutions of Maxwells equations.

The background with non radiation is that at the time of development of the theory of the atom, there was a mystery why the electron did not radiate as it accelerates in it's orbit. Today the answer is another theory, called quantum mechanics, that describe the physics of the electron and presents source terms that are not depending on time and hence does not radiate: see, nonradiation. But before quantum mechanics was produced, scientists tried to use maxwells equations and electrodynamics to yield solutions that does not radiate. During the second half of the 1900 centruary there was some development on characterisatin of non radiating electromagnetic fields and Haus derived the condition that a charge field does not radiate the fourier transform of the charge field does not have any components associated with the speed of light. During the 1970-ies the american military seam to know about solutions that does not radiate and Randel Mills, in his theory, GUTCP, took that solution further and assumed it as a basis for electron orbitals that makes up the atoms and ions. I have discussed his theory quite a lot and the opponents seam to not admit anything that Mills bring out as of value. But we could ask ourselves does these solutoin radiate or not. If you look at the first proof of Mills you will se a fantastic complex deduction which I believe is wrong. The thing is that you can deduce it much simpler and I will show you the steps. In any case I would be surprised that a model of electromagnetic fields, that has all properties to enable a model of an atom and do not radiate, should not have a bearing in the construction of the atom. But enough pladder, lets jump to math. The solution mills proposes looks like

$$\rho = A_{lm} Y_{lm}exp(iwt)\delta_{r=r_0}$$

with $$A_{lm}$$ constants and $$Y_{lm}$$ the spherical harmonics and $$delta$$ the uniform measure of a spherical shell at distance $$r_0$$. Now using the Haus theorem we calculate the fourier component:

$$F(\rho) = A_{lm}\int Y_{lm}exp(iwt)\delta_{r=r_0}exp(iut + i k\cdot \hat r)d\hat r$$

The trick is to take express $$exp(ik\cdot \hat r)$$ as an expansion of spherical harmonics. It turns out that we have the identity: expansion

$$exp(i k\cdot \hat r) = \sum_{lm} C_{lm}j_l(|k||r|)Y_{lm}(k)Y_{lm}(r)$$

, with $$C_{lm}$$ nonzero constants.

So putting this in and restrict the integration to the spherical shell and also do the integration in time domain we geta. Let's also interchange the sum and integral:

$$F(\rho) = \sum_{l'm'}C_{l'm'}A_{lm}j_l'(|k|r_0)(\int_{r=r_0}Y_{l'm'}(r)Y_{l'm'}(k)Y_{l,m}dS) \delta_{u=w}$$

No this integral is zero unless $$l=l',m=m'$$ due to the orthogonality of the spherical harmonics and we get:

$$F(\rho) = C_{lm}A_{lm}D_{lm} j_l(|k|r_0) Y_{lm}(k)\delta_{u=w}$$

, with $$D_{lm}$$ the norm of the spherical harmonics.

So the time fourier component $$u$$ is only $$w$$ and hence for a wavelike packet we must have $$|k| = c|w|$$ and if we take $$r_0 c w$$ a zero to the bessel function, we find that all light like wave numbers lead to a zero fourier component, hence according to Haus theorem for specific $$r_0,w$$ the distribution does not radiate and we're done.