Planck's Natural Units

Planck established units of length, mass, time and temperature utilizing the two constants, h and k, from 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".

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

These values are slightly different 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.

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.

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

(4)

(5)

This also fits Einstein's equation and Planck's radiation equation.

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)

So 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 atom 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. 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 has ever been derived.

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". This has now been accomplished for the" mean of a probability distribution that is represented by the energy state equation" and changes in state.

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 the natural units for frequency and energy. This analysis leads to "the meaning of temperature", as we shall see in a future development.