The Compton Wavelength Represents the
Fractal Reciprocal of the Planck Implied Length

[Natural Unit of Length, Planck Implied Time and the Velocity of Light in a Vacuum]

by Charles William Johnson

In this brief commentary, I illustrate that the Compton wavelength is a reciprocal fractal expression of the Planck implied length value. I also show why the Planck implied time is significant in relation to the natural unit of length as given by the CODATA, which represents a case in favor of its inclusion in the CODATA listing.

The constant of Planck time [5.39124] is derived as the Planck implied time [8.637717964] divided by the elementary charge [1.602176487]:

8.637717964 divided by 1.602176487 equals 5.39124.

In a previous essay, I discussed how the Planck implied time is derived as of Planck time by multiplying Planck by the elementary charge. In this case, one must simply reverse the order of computational events.

Also, in another essay, I have suggested including the Planck implied constant values in the CODATA recommended values for the fundamental physical constants. Consult the Homepage of my web-site, www.earthmatrix.com.

In another essay about the use of metric time, I have shown the significance of the 1.1574074074 conversion factor. The 1.1574074074 fractal expression derives from the importance of the reciprocal of 86400 seconds in an Earth day according to the 24-hour clock standard. In a sense, then, the 1.1574074 conversion factor is an Earth-bound category, just as is the reciprocal of the 86400-second clock standard.
[http://www.earthmatrix.com/sciencetoday/metric_time_constants.pdf]

The Planck implied time constant, 8.637717964, then, is a modified derivative of the 864 expression by way of Planck time [5.39124] and the elementary charge [1.602176487].

5.39124 times 1.602176487 equals 8.637717964

The reciprocal, then, of Planck implied time is 1.15771319, similar to the ideal conversion factor for metric time [1.1574074074].

Enter the CODATA natural unit of length given in the symbolic formula of h-bar divided by electron mass times speed of light in a vacuum: 3.8615926459. [Again, as customary, I am using fractal multiple expressions with a floating decimal place instead of the generally accepted scientific notation.] The natural unit of length is but the fractal numerical expression of the Compton wavelength. Consider.

"The Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons (a process known as Compton scattering). The Compton wavelength of a particle is equivalent to the wavelength of a photon whose energy is the same as the rest-mass energy of the particle. The Compton wavelength, ?, of a particle is given by:
Lambda = h divided by mc
where h is the Planck constant, m is the particle's rest mass, and c is the speed of light. The significance of this formula is shown in the derivation of the Compton shift formula.
The CODATA 2006 value for the Compton wavelength of the electron is 2.4263102175±33x10^-12 meters.[1] Other particles have different Compton wavelengths.. When the Compton wavelength is divided by two pi, one obtains a smaller or "reduced" Compton wavelength:
Lambda-bar times speed of light in a vacuum The reduced Compton wavelength is a natural representation for mass on the quantum scale, and as such, it appears in many of the fundamental equations of quantum mechanics." 386.15926459 x 10^-15 m. - [Source: www.wikipedia.com]

2.4263102175 divided by 2-pi equals 3.8615926459 fractal

The Compton wavelength numerical value is presented also in the CODATA as the natural unit of length: fractal 3.8615926459. Now, consider its relationship to the Planck implied time value:

3.8615926459 times 8.637717964 equals 3.335534816.

[n.u. of length | Planck implied time | reciprocal of speed of light]

The natural unit of length, i.e., the Compton wavelength, times the Planck implied time yields the fractal reciprocal of the speed of light in a vacuum.

The product, 3.335534816, is suggestive of the reciprocal of the speed of light in a vacuum 3.33564095. This particular equation makes all the sense in the world; it just needs to be turned around.

Reciprocal speed of light | Planck implied time | n.u. length

3.33564095 divided by 8.637717964 equals 3.86171552

It is similar to stating that an airplane that flies at a speed of 3.33564095 motion units within 8.637717964 time units will travel 3.86171552 natural units of distance measurement ---so to speak. The natural unit of length traveled in Planck's implied temporal fraction of a second traveled at the cited fractal reciprocal speed of light in a vacuum.

The natural unit of time given by the CODATA, 1.2880886570 represents a multiple of the base equation shown in the previous computation as it involves the square of the speed of light in its terms: h-bar divided by the electron mass times the square of the speed of light. In this sense, the natural unit of time given by the CODATA is yet a reduced fractal multiple of the 8.637717964 Planck implied time constant. In order to reduce this particular value even more one could cube the speed of light in vaco for the natural unit of time; and so on infinitely so.

Setting aside any considerations for the CODATA natural unit of time, 1.2880886570, it is necessary to consider the reciprocal of the derived value 3.86171552 [instead of the CODATA value 3.8615926459 for the natural unit of length].

Consider then the reciprocal of the 3.86171552 revised numerical value.

1 divided by 3.86171552 equals 2.5895227

The reciprocal of the Compton wavelength is the fractal expression of the Planck implied length.

Previously, I have shown how the Planck implied length may be considered to be 2.589520951 fractal numerical value. Variations for this value occur as of the distinct fractal numerical values of the terms chosen for its derivation. The Planck implied length is derived as of Planck length, 1.616252 and the elementary charge.

1.616252 times 1.602176487 equals 2.589520951.

This fractal numerical implied value is close, but not exactly the same, as the reciprocal value derived from the natural unit of length [2.5895227]; the difference is fractal 0.000001749.

The previous computations mean that the Planck constants for length [1.616252] and for time [5.39124] require adjustment in order to bring all of the Planck implied values into numerical correspondence with one another; i.e., a 2.58952+ value would derive for each of these particular constants as illustrated above. The obvious conclusion is, if the Planck implied values are in non-correspondence then the resulting reduced Planck constants [called the Planck constants by the CODATA] are themselves in non-correspondence. This means that they need to be empirically re-measured once the theoretical interrelationships are laid bare.

In a previous essay, different pairs of fundamental physical constants produce a 2.58960521 numerical value for the Planck implied length.

1 divided by 3.8615926459 equals 2.5896051

Those equations would have to be taken into consideration as well in order to bring all of those fundamental physical constants into correspondence with a single value for Planck implied length [in the range of fractal 2.5895 - 2.5896+]. As it stands, the different CODATA fundamental physical constants have numerical values that produce different expressions for the same Planck implied constant; a 2.5895+ value may apply to one set of constants, while a 2.5896+ value may apply to others.

By bringing the realm of the Planck implied values into the analysis one would be able to adjust the Planck constants and some of the numerical values of the fundamental physical constants to where they would all be relational to one, employing the same value for any given constant. As it stands, the CODATA numerical values employ different values for the same implied constant regarding different pairs of fundamental physical constants. When considering any pair of fundamental physical constants [Planck or non-Planck constants] any given implied constant should have the same fractal numerical value, even though its unit of measurement may be different.

The self-same numerical value for different pairs of fundamental physical constants obtains as of the self-same coordinates of all spacetime/motion events of the different forms of matter-energy. Inasmuch as all spacetime/motion events are relational, their fractal numerical expressions will also be as such, for all aspects/levels of space, for all moments/processes of time, and for all relations/systems of motion.

In summary, a natural unit of mass would be reflective of all aspects/levels of space. A natural unit of time would be reflective of all moments/processes of time. A natural unit of motion would be reflective of all relations/systems of movement. The different natural units of measurement would necessarily carry over from one event/realm of spacetime/motion into all events/realms of spacetime/motion. All constant ratios of the natural units of spacetime/motion would necessarily have to share the same numerical values for each chosen relationship of compared events.

In other words, a particular Planck implied constant value would necessarily have to be the same for any spacetime/motion event being analyzed, as demonstrated in this brief essay. Summarily, the Compton wavelength or the natural unit of length must necessarily reflect the reciprocal numerical fractal value of the Planck implied length. And, these physical numerical values must be the same for all comparative purposes with regard to the other fundamental physical constants. Any variation in the fractal numerical expressions means that the measurements are approximate.

Earth/matriX Editions New Orleans, Louisiana March, 2010 Copyrighted by Charles William Johnson. All rights reserved. Reproduction prohibited.

THE PLANCK CONSTANTS BASED ON THE FUNDAMENTAL PHYSICAL CONSTANTS AND ERRORS AND OMISSIONS OF THE CODATA The author derives the Planck constants as of the fundamental physical constants, instead of as of the traditional symbolic formulae given in the CODATA. The implied numerical values that are generally omitted in the science literature and which serve as a basis for deriving the Planck constants are identified in detail in their basic math. As of the analysis of the implied Planck values, errors and omissions of the CODATA Planck values are identified and treated extensively. The Planck constants are reverse engineered in order to employ the fundamental physical constants as their computational foundation. Various Earth/matriX Tables of the Planck Constants Based on the Fundamental Physical Constants are presented for the first time, illustrating how the fundamental physical constants serve as the foundation to the Planck constants. Purchase and download the complete e-book: Errors and Omissions in the CODATA and The Planck Constants Based on the Fundamental Physical Constants PDF file 142 Pages Earth/matriX Editions ISBN 1-58616-463-5 Price: $8.19US Introductory Offer Earth/matriX SCIENCE TODAY The Planck Constants Based on the Fundamental Physical Constants, With Tables. ISBN 1-58616-464-3. By Charles William Johnson. Earth/matriX Editions