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It seems intuitive that static electricity would have the unit electric charge of an electron as the basic unit of charge (see Link).  In other words all charge values should be integer multiples of the charge on the electron.  This is true, but there is another fundamental unit of charge known as the Planck charge that needs to be explained. The Planck charge is 11.7 times larger than the charge on the electron. The explanation of this anomaly will help explain how the quantum world works.  And it will give some support to the concept of λ-hopping.

The Foundation of a Quantized Electro-Static Force:

1. Can we make the electro-static force (Force = keq1q2/d2) fit with the Planck-Einstein equation?
Note that Coulomb's constant ke = 1/(4πe0).  Where e0 is the permittivity of free space.

The Planck-Einstein equation is the starting point of quantum theory and is given by the equation:
Energy = hf = Nhc/λ, where N is the number of photons and hc/λ is the energy of each photon.

2. As a first step we can take the quantized energy equation [Energy = Nhc/λ] and convert it to something more natural in handling classical distances.  To do this we will say that wavelength λ of each quanta (hc/λ) is the distance d that separates things like charged spheres. With this assumption we now have that the electrostatic energy is E = Nhc/d. Let’s keep going an see if these photons makes sense. Please note that these photons are like all the photons in this presentation. They are λ-hoppers. They consist of a point like essence and a wavelength over which they hop (leave space-time). We can consider wavelength just as an ordinary distance.

3. We can convert the energy E = Nhc/d to force by dividing by d to get F = Nhc/d2.  This results from the fact that energy is equal to force times distance and F = E/d.

4. We now have two equations for the force between charges (F = keq1q2/d2) and (F = Nhc/d2). We can solve for N and get:   N = q1q2/(4πe0 hc).  This is interesting, but we can go further.

5. Noting that the Planck charge squared (Pc2) is equal to (4πε0)hc, we can take N = q1q2/(4πe0 hc) and rearrange things to get N = q1q2/Pc2 as the number of photons connecting two charged objects q1 and q2For more information on the Planck Charge (and the relation Pc2 = (4πε0)hc) click here.

Note that for every Planck charge in q1 and q2 there is an electro-static photon connecting themAnd as with the case of gravity (see section 17), I wish I had an easy visualization of why this is true.  However, it just fell out of the equations.

6. This is very interesting: If the number of photons N that connect two charges is chosen to be        q1q2/Pc2, then the classical force between static charges (but now quantized) results.

7. However there is something funny about the result N = q1q2/Pc2.  We would think that if we had two electron charges for q1 and q2 we would get N=1, but no, instead we get N = 0.00729735.
Thus we cannot say that Planck charge is the quantum unit of charge.

8. We can patch this in an interesting way. Take N and multiply it by 1/α where α is the fine structure constant and the result will be N=1.  The Calculation is N = 1 =(1/α)(e2/Pc2), and α = e2/Pc2 . If we make this calculation we get α = 137.03598939. This compares with the Codata measured value of 137.035999139, not too bad. I believe it is not spot on due to inaccuracies in the value for the Planck charge. See link for more information on the fine structure constant.

9. Can we make physical sense of the above numerology?

Here is the answer (and it is the whole premise of this website): The electron is a quantum mechanical particle and as such it cannot be treated as a continuous entity. The electron moves by λ-hopping. It exists in space-time for awhile and then disappears for awhile (see section 30 for a diagram of this).

Sommerfeld who first considered the coupling constant in 1916 (see link) defined it as the velocity of the electron in the lowest orbital of a hydrogen atom as compared to the speed of light c. I think of this electron in the lowest energy orbit as a thing that λ-hops back and forth about the proton. It hops to one side of the proton and stops for a bit then it hops to the other side of the proton.

The speed of the electron in a hydrogen atom is 2.188 x 106 m/second (see link). This is slower than the speed of light (3 x 108 m/second) by a factor of 137..... interesting.

And there are probably different coupling constants for each energy level in the hydrogen atom, and for each of the other elements and molecules. And yes the coupling constant is a useful concept and we can see that it can vary depending on orbitals and other conditions. The standards body NIST uses the quantum hall effect to measure α. Note that α  is equal to (1/137)1/2 = 11.7.

But why did anyone choose the Planck Charge (4πε0ħc)1/2 as a Planck Unit instead of [α(4πε0ħc)]1/2 ?
It would seem obvious that the Planck charge should be the charge on an electron.

Here is what the Planck charge would look like if it were the electron charge:

Planck charge = e = [α(4πε0ħc)]1/2   as opposed to Planck charge = 11.7e =(4πε0ħc)1/2 .

Note 1: It is unfortunate but the factor of α (the coupling constant) in the above definition is a
stumbling block. No one will agree that it is a universal constant in the sense that G, h, or c is
a universal constant. And thus it is not allowed in the Planck units and thus we are stuck with
the Planck charge being 11.7 times to big.

Note 2: The charge on the electron is also a measured value that can also be calculated as shown by
the above equation.

Note 3: The coupling constant
α is a strong indicator of the validity of λ-hopping as an appearance/
disappearance phenomena for particles.

Note 4: The definition of the Planck charge should be:  Planck charge = e = [α(4πε0ħc)]1/2