A derivation of Planck's constant

v. 6 n. 27

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Planck's constant, h, became apparent from an experiment with blackbody radiation and experimentally confirmed in other areas of physics. It is the basis of quantum mechanics and not normally derived. Any theoretical derivation necessarily excludes quantum mechanics.

The following is from classical considerations -- special relativity and Newton's gravity -- and might suggest that Planck's constant and quantum mechanics are not fundamental, rather based on the relativity constant, c, and gravitational constant, G. Also, it has been discussed that G and the Hubble constant, H_o, are complementary, [1] so that, all told, Planck's constant may be based on large-scale considerations, entirely changing the focus from essentially subatomic to essentially cosmic.

Special relativity indicates that the mass-energy of an entity increases with increasing velocity; let

E ? x c^2 [(1- v^2/c^2)^-1/2 -1] .................. (v→c) ......................... (1)

where E is relativistic kinetic energy and defined as the energy of a "spacetime wave" or particle precursor to distinguish it from a gravitational wave where v=c; parameter x is to be identified (normally considered a rest mass in such an equation). Recall the discussion where accommodations are made in spacetime instead of an increase in mass when considering particles increasing in velocity. [2]

Special relativity also indicates that an entity contracts with increasing velocity, let

λ ? λo (1- v^2/c^2)^1/2 ................................ (v→c) ........................... (2)

where λ is defined as the wavelength of a "spacetime wave" and λo reads λ sub-0, to distinguish λ from a gravitational wave where v=c. Spacetime wave frequency would be

ν = v / λ ................................................................. (v→c) ............................ (3)

(note the distinction between Greek letter ν and velocity v). Substituting Equation (2) into Equation (3) indicates that frequency increases with increasing velocity, so that energy appears directly related to frequency,

E = k ν ........................................................................................................... (4)

where E is defined in Equation (1) and k is the constant of proportionality. Then

λ = k / xv. .............................................................. (v→c) .......................... (5)

Since the situation is near light speed here there could be sufficient relativistic kinetic energy to produce a non-zero rest mass particle. It might be suspected that λ is a de Broglie wavelength, if k is Planck's constant, h, and x is at least a rest mass.

The following equation has been discussed in various situations,

m/r^2 = A/G ............................................................................................... (6)

where in the initial derivation r is the radius of the local galactic supercluster containing mass m and A≈10^-14 m/s^2 is the acceleration of this supercluster; Equation (6) is shown to be scale invariant. [3] Let m=x, r=k/xv from Equation (5). Then

k = (G/A x^3 v^2)^1/2 ≈ h ............................ (v→c) ......................... (7)

when x is the mass of a light quark. Planck's constant, then, stems from stable fundamental matter particles in conjunction with the large-scale parameters, G, A and c. That it is also central to the photon and light,

E = hν ........................................................................................................... (4a)

makes it a bridge between matter and light, and large and small scales.

[1] Complementarity of the Hubble and gravitational constants | LinkedIn

[2] (3) Views of the fourth dimension, stressed space | LinkedIn

[3] (3) An explanation of dark matter and dark energy from unmodified Newtonian gravity* | LinkedIn

Cover image: https://edulab.com/plancks-constant-what-is-it-and-why-does-it-matter/