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Planck's Natural Units
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Planck's Natural Units

Planck established basic units of length, mass, time and temperature utilizing the two constants, h and k, using the universal law of radiation plus the speed of light and the gravitational constant. He claimed that these four units are independent of special bodies or substances and which "retain their significance for all times and environments" and described them as "natural units". Later on you will see how I was able to eliminate one of these constants!

The numerical values of the four basic constants of physics (Planck, 1913) are:

These values than those of modern physics. Planck's natural units are:

Planck claimed that these natural units would always remain the same as long as the law of gravitation, the speed of light in a vacuum, and the principles of thermodynamics remain valid. The values of some of the constants have changed slightly (see SI units), but Planck's original units will be used.

The form of above equations for the natural units seemed strange to me when I first saw them. The strangeness is due to the inclusion of the gravitational constant, which produces the square root function, so let's see if we can get rid of it:

These equations can be manipulated into the more recognizable form of ratios. First consider the product of mass times length:

(1)
(2)

(3)

which fits Einstein's energy equation and Planck's radiation equation but without the G-factor.

Now we will do the same for the product of mass and time:

(4)

(5)

This also fits Einstein's equation and Planck's radiation equation without the G-factor.

Now we multiply the temperature unit T by Boltzmann's constant k,

(6)

and then divide this result by the time unit:

. (7)

This is a very interesting result, since the energy equation is now

. ,(8)

and again without the G-factor. So who needs the G-factor? What is the meaning associated with this result? The answer to this question is obtained from Planck's analysis of the energy state of an atom, which he called an "oscillator" (see "Planck's Columbia Lectures", Chapter 6), and a new analysis of his energy state equation ("Planck's Columbia Lectures", Chapter 10). In brief terms, the resulting conclusion is that (hf = kT ) always occurs at the half-power point of the spectrum of the Planck's energy state equation, regardless of the temperature. In my analysis of Planck's radiation theory, I recognized that the energy state equation relates to a set of time functions, irregularly spaced (random). Utilizing the methods of transform analysis, the time function was derived, and this conforms to the "universal quantity of action" for "impact actions" and changes in the state of an atom. To my knowledge, this is the only "quantum time function" that has ever been derived (a quantum of energy w.r.t. time)!

Planck asserted that "to show in detail and in what time intervals the result [of impact actions] is arrived at will be the problem of a future theory", and that it "must give a closer explanation of the physical significance of the universal elementary quantity of action, a significance which is certainly not second in importance to that of the elementary quantity of electricity". The above result for the quantum of electrical energy, was derived using the" mean of a probability distribution that is represented by the energy state equation" and changes in state (illustrated in Chapter 10).

The Planck normal frequency unit is obtained by inverting the time unit,

. (9)

This frequency unit can then be inserted into equation (8),

(10)

which is Planck's natural unit for energy. Note that this equation works well for the parameters hf and kT. As an exercise, substitute the parameters of modern physics, which are slightly different, and see how well they equate. Similarly,

, (11)

and we now have determined all of the above natural units for frequency and energy without any G-factor, and the constants of frequency and wavelength have also been determined. This analysis leads to "the meaning of temperature", as we shall see in a future development.

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