Natural Science Forum / Physics / General Physics / July 2008

This article makes rigorous definitions of a few constants,
and discusses what physical properties should be considered invariant.

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. Assume that  K = a universal distance per mass constant.
  K = 1.0585382 x 10^13 meters per kilogram for E-M interactions.

6. Assume that fine structure(E) = ( M(P) * K / L(C) ) ^1/3
then    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 simplest 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

Comments:

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
  accommodate change.

 The simplest system would consider the rest masses
  to be constant, the distance common to the masses L(C) to be an
  independent variable and all properties to be dependent variables.
  Note that the distance L(C) is related to the common period of the system.

3. Schrödinger's Equation would be symmetrical to both the electron and the
  proton if 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, but the equation does not provide a symmetrical look at the
  classical system absorbing or emitting the frequencies. Schrödinger'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.

  The more fundamental approach would be to use the time intervals
   associated with the distances, as spaces are fundamentally time intervals.