This article makes rigorous definitions of Planck's constant and the fine structure constant and suggests that they are not in fact constants.
Let us consider a system composed of one electron and one proton.
1. Let M(P) = the mass of the proton. 2. Let M(E) the mass of the electron. 3. Let c = a universal distance per time constant. ( The speed of light. )
4. Two bodies interact about a common point in a common time. The common point is the center of mass of the system and the common time is the period of the system.
Let T(c) = the common period divided by 2 times pi = L(c) / c
where L(c) is the distance light travels during one radian of interaction of the electron-proton system.
5. Let K = Potter's electro-magnetic universal distance per mass constant. K = 1.0585382 x 10^13 meters per kilogram for E-M interactions.
6. Then the fine structure(E) = ( M(P) * K / L(c) ) ^1/3 and fine structure(E)^0 * L(c) = 1 / ( 2 * Rydberg constant ) and fine structure(E)^1 * L(c) = 2 * pi * Bohr Radius and fine structure(E)^2 * L(c) = compton's wavelength and fine structure(E)^3 * L(c) = 2 * pi * classical electron radius = M(P) * K
As interactions are symmetrical about the common center of mass, we can define a fine structure constant for the **proton** and obtain the following equations:
fine structure(P) = ( M(E) * K / L(c) ) ^1/3 fine structure(P)^0 * L(c) = 1 / ( 2 * Rydberg constant ) fine structure(P)^1 * L(c) = 2 * pi * Bohr Radius(proton) fine structure(P)^2 * L(c) = compton's wavelength(proton) fine structure(P)^3 * L(c) = 2 * pi * classical radius(proton) = M(E) * K fine structure(P)^3 * M(P) = fine structure(E)^3 * M(E)
7. Let h(E) be the Planck's constant for an electron.
8. Let h(P) be the Planck's constant for a **proton**.
Note that: M(E) * M(P) * K^2 = fine(E)^3 * fine(P)^3 * L(c)^2 = h(E) * fine(P) * K / c = h(P) * fine(P) * K / c
Also note that: h(E) * K / c = fine(P)^3 * fine(E)^2 * L(c)^2 = M(E) * K * fine(E)^2 * L(c)
and symmetrically: h(P) * K / c = fine(E)^3 * fine(P)^2 * L(c)^2 = M(P) * K * fine(P)^2 * L(c)
Equations showing the simpliest relationships between Planck's constant and the Fine structure constant: fine(P) * h(P) = M(P) * M(E) * K * c fine(E) * h(E) = M(P) * M(E) * K * c
Note: As K and c are universal constants, and as we are considering rest masses to be constant, h(X) and fine(X) must vary reciprocally when a system such as a hydrogen atom is changing states.
The relationship between the orbital velocity of a body and the fine structure constant is: sine(X) = velocity(X) / c = fine(X) * charge ratio
1. The common period is associated with Rydberg's constant. In other words, the distance symmetrical to both bodies is the reciprocal of Rydberg's constant. The other distances ( comptons wavelength, etc. ) relate to a particular body.
2. If we assume that rest masses are constants, we have to acknowledge that the h's and fine structure constants must vary for a system to accomodate change. The simpliest system would consider the rest masses to be constant, the distance common to the masses L(c) to be an independent variable and all other properties to be dependent variables.
Note that the distance L(c) is related to the common period of the system.
3. Schrodinger's Equation would be symmetrical to both the electron and the proton if it were based on the mass products rather than a "constant" associated with only one of the bodies. The equation works because the incoming and outgoing frequencies are common to both parties to an interaction. Schrodinger's Equation, like Planck's constant is biased in favor of the electron.
4. I emphasized distances, rather than more fundamental times and angular displacements, in order to more clearly show the relationships between the common physical constants.
5. Observe that the foregoing is for a one electro/one proton system, and the ELECTRO-MAGNETIC shape of these particles would determine how the Exclusion Principle comes into play.
6. The fundamental unit of reality is a cycle, and Planck's constant for the electron equates the radius of an electron cycle to a unit of electron ACTION, and Planck's constant for the proton equates the radius of an proton cycle to a unit of proton ACTION,