A NEW DERIVATION OF PLANCK’S CONSTANT To presentday physical science the numerical value of Planck’s constant is a mystery: quantum mechanica does not have a theoretical method for its calculation. By contrast the Reciprocal System of theory derives the value of all physical constants, including Planck’s constant, from its fundamental postulates. However, because of errors in the previous derivations, this paper presents a new, dimensionally sound method for the calculation. Larson¹ was the first to attempt to derive Planck’s constant from the Reciprocal System. Because of the change in the calculated natural values of mass and energy in the second edition of his work², the original derivation has been invalidated. The factor of three that was used is dimensionally incorrect since the photon is a onedimensional vibration. And the use of the cgs gravitational constant in such an equation is wrong since the result cannot be converted to a different system of units such as the Sl (mks) system. The remainder of Larson’s original equation (including the use of the interregional ratio and the square of the natural unit of time) will be shown to be correct. Nehru³ made the second attempt to calculate the constant. However, he started by setting the dimensions of energy to be space divided by time, which is, of course, the reverse of what they are. The rest of the derivation was very tortuous, although he ended up with a good numerical result (with faulty dimensions). Let us now proceed with the new derivation. First, consider conceptually the linear vibration of the photon. The oscillation takes place over one space unitwhich, simultaneously, is also one time unit. In the material sector of the universe, we define frequency to be cycles/sec, because here it is time that appears to have a uniform progression; in the cosmic sector of the universe, hypothetical cosmic observers would define frequency to be cycles/cm (or some such length unit), because there it is space that would appear to have a uniform progression. Actually, the photon exists at the boundary between the two sectors, where both space and time progress uniformly. Here the correct, nAtural definition of frequency must be cycles/(cmsec) (or equivalent units). To put it another way, frequency in the natural sense is the number of cycles per spacetime unit. Photons of all frequencies can be observed in both sectors, and the only way that this could be possible is if the denominator of the natural definition contains both a space unit and a time unit. This then causes Planck's constant to have the actual dimensions of ergcmsec. However, if the dimensions of frequency are assumed to be cycles/sec, rather than cycles/(cmsec), then the dimensions of Planck’s constant are ergsec. Let E be photon energy, h be Planck’s constant, and Öbe photon frequency. Then, as usual, we have
In spacetime terms, equation (1) is
In the cgs system of units, equation (1) is
Observe, in both cases, the dimensional consistency. Since the oscillation of the photon takes place with in a unit of spacetime, the interregional ratio must be contained within Planck’s constant. With this factor and the dimensional information from above, Planck’s constant is
where t is the natural unit of time (1.620666 * 10'18 sec). Ref. 3 states that the ratio of (sec/cm)/erg is 2.236066 x 10^{8}. This figure is deduced as follows. Dimensionally unit mass is t³/s³ , or 3.711381 * 10^{32} sec³/cm³. Avogadro’s constant is the number of atoms per gram atomic weight. 6.02486 * 10^{23}. The reciprocal of this number, 1.66979 * 10^{24}, in grams, is therefore the mass equivalent of unit atomic weightz². Thus to convert from the unit sec³/cm³ to grams we must divide by 2.236066 * 10^{8}. From the euprsssion E = mc² we see that the sa.me conversion factor must apply to energy (in ergs) to keep the equation balanced. (Nehru³ modi ied his equation to include secondary mass; however, his rasulting equation is dimensionally incorrsct. Furthermore, secondary mass varies between the subatoms and atoms and so cannot be a part of the conversion factor). Thus the numerical value of Planck’s constant is
(when frequency is assumed to have the dimensions cycles/sec). References
