Both the Faraday law and Ampere/Maxwell law are partial differential equations as is the Schrodinger and Dirac Equations. They are linear partial differential equations.
So how is it that the Schrodinger and Dirac Equations are derived from the Faraday and Ampere/Maxwell laws? How are they derived? It is not as difficult as one would imagine.
If you look on page 73 or 76 of The Elements Beyond Uranium, 1990, Seaborg & Loveland you will see pictorials of the shapes of orbitals of atoms. The shapes are with a proton nucleus and a electron orbital.
Now if you take a single transverse wave of this: E | |___ B
And project that wave from 2nd dimension into 3rd dimension, the most you can get out of the wave is a 3rd dimensional trough like figure. You get a U shape projected into 3rd dimension as a trough, like a half open pipeline.
However, if you have a double-transverse wave:
the destructive-interference of the wavefronts forms a closed loop, a circle or ellipse shaped closed loop and if you project a circle into 3rd dimension you can end up with a sphere or a ellipse shaped lobe figure.
So the Schrodinger and Dirac Equation as pictured in that book are figures that are closed loops of the Maxell Equations projected into 3rd dimension.
Now it even talks about how the Dirac Equation makes the p orbitals more donut shaped than what our chemistry textbooks pictured as p lobe shaped.
So in essence, given the Maxwell Equations, they insist on a closed loop which is a double transverse wave. And once we are armed with a double transverse wave we apply the Maxwell Equations again on the double transverse wave by projecting those closed loops of 2nd dimension into 3rd dimension and we end up with spheres, ellipsoids, elongated ellipsoids and those figures shown in the book above.
So the Schrodinger equation and Dirac equation are just simply the application of the Maxwell Equations twice.
We see it also in the mathematical form of the Maxwell Equations with monopoles of the Faraday law and the Schrodinger Equation, which if we replace the symbolism, we end up with one and the same thing.
But the easiest way of describing the mathematical equivalency is by projection of 2nd dimension closed loops into 3rd dimension, which is what the Schrodinger/Dirac Equations are as a projection of the Maxwell Equations.
I wish it could have been more complicated, and I do recall the comment that Feynman made a long time ago about Schrodinger, and I honestly do not understand the low opinion that Feynman had towards Schrodinger, perhaps envy, but Feynman claimed that Schrodinger got his equation out of thin air and had no way of deriving it.
But above, I show Feynman (posthumously) how the Schrodinger and Dirac Equation are derived from purely the Maxwell Equations.
Google's New-Newsgroups censors AP posts and halted a proper archiving of author, but Drexel's Math Forum does not and my posts?in archive form is seen here: