﻿ The Magnitude of Planck's Constant and Its Significance
San José State University

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Thayer Watkins
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The Magnitude of Planck's
Constant and Its Significance

Planck's constant h is often considered a fundamental parameter of the Universe. Its value in the MKS (meter-kilogram-second) system is 6.626×10−34 joule-sec. The notable fact is that Planck's constant is dimensional and hence its magnitude depends upon the system of units used to express it. In the cgs (centimeter-gram-second) system it is 6.626×10−29 erg=seconds. But energy can be measured in calories, electron volts, British Thermal Units, kilowatt-hours just as well as in joules and ergs. Also time can be expressed in minutes, hours, days, years, centuries, millenia and more practically in milliseconds, microseconds and nano seconds. Some of these combinations will make Planck's constant a large number. Thus one cannot say whether Planck's constant is a small number or not. It is obviously not a fundamental parameter of the Universe. It is a crucial parameter which depends upon the dimensional units used.

The crucial parameter could be Planck's constant divided by 2π, what is called h-bar, h. What appears in the Uncertainty Principle is ½h. Thus the product of the uncertainty in the position and the uncertainty in the momentum of a particle must be greater than or equal to ½h. In a system in which the unit of mass is the rest mass of the proton (1.672×10−27 kilograms, the unit of velocity is the speed of light in a vacuum (2.998×108 meters per second, and the unit of length is the scale parameter for the nucleus based upon the Yukawa relation (1.522×10−15 meters) the product of the uncertainties in position and momentum must be greater than or equal to 0.06914. That is to say, Planck's constant would have a magnitude of 0.86886=2(2π)(0.06914).

A crucial parameter of the universe cannot be a dimensioned number; it must be a dimensionless number. A prime candidate for such a parameter is the so-called fine structure constant, which is approximately 1/137. Its value is the ratio of the constant for the electrostatic force to the product of h-bar and the speed of light c. Of course, many dimensionless constants can be constructed from Planck's constant and other physical measurements. And also of course, any function of a dimensionless ratio of physical constants, such as a square or multiplication by a dimensionless constant, is also a dimensionless number.

Consider the following construction. Divided the product of the rest mass of the proton and the speed of light into Planck's constant. The result is 1.321854×10−15 meters. There is a scale parameter s0 based upon the Yukawa relation and the mass of the π-meson which is 1.522×10−15 meters. The ratio of these two distances is the value 0.86886, a dimensionless quantity. However this number is completely independent of the value of Planck's constant. Yukawa relation is

#### s0 = h/(mπc)

where mπ is the mass of the π-mesons. Note that the above formula involves h-bar rather than h. Thus

#### [h/(mpc)]/s0 = [h/(mpc)]/[h/c)] = 2π(mπ/(mπ) = 2π(0.14925) = 0.9378.

It just happens that the ratio of the mass of the proton to the mass of the π-meson is approximately 2π.

The dimensionless quantity that could be considered crucial is one such that the mathematical stability of important physical quantities depends upon the magnitude of its value.

## The Fine Structure Constant and Other Ratios of Force Constants to hc

One of the most famous dimensionless constants is the fine structure constant. This is the ratio of the constant in the formula for electrostatic force to hc where h is Planck's constant divided by 2π and c is the speed of light in a vacuum.

Coulomb's Law of Electrostatics is that the force between two charges q1 and q2 separated by a distance r is given by

#### F = (1/(4πε0))q1q2/r²

where (1/(4πε0)) is a constant equal to 9×109 kg*m3/s2. The quantity ε0 is known as the permittivity of free space.

The charge of any body is essentially equal to the net number of elementary charges it contains times the value of the elementary charge; qi=qeni. Thus the force formula could be represented as

#### F = (1/(4πε))qe²n1n2/r²

Thus the force constant in units of kg*m3/s2 is (1/(4πε))qe²=(9×109)(1.60218x10-19)² = 2.3103×10-28 kg*m3/s2. The ratio of this constant to hc=3.1616×10-26 kg*m3/s2 is 7.34844×10-3 or approximately 1/37.06.

## The Gravitational Force

Newton's Law of Gravitation is that the force between masses m1 and m2 separated by a distance r is given by

#### F = Gm1m2/r²

where G is a constant equal to 6.67259×10-11 m3/kg.

The mass of any body is essentially equal to the number of nucleons it contains times the mass of a nucleon; mi=mnni. Thus the force formula could be represented as

#### F = Gmn²n1n2/r²

Thus the force constant in units of kg*m3/s2 is Gmn² = (6.67259×10-11)(1.6749×10-27)2 =1.871855×10-64 kg*m3/s2.

The ratio of this constant to the product of h-bar and the speed of light in a vacuum, hc=3.1616×10-26 kg*m3/s2, is 3.76915×10-39. This is the coupling constant for the gravitational field.

## The Nuclear Force

The force between two nucleons may be given by the formula

#### F = H*e−λr/r²

The justification for this formula is that the nuclear force is carried by particles subject to decay; i.e., the π mesons. The population of remaining particles is a negative exponential function the time since emission which translates into a negative exponential function of distance. These remaining particles are spread over an area of 4πr². The intensity is thus proportional to e−λr/r². For more on this model see Nuclear Force.

An estimate of H based upon the separation distance of the nucleons in a deuteron being 3.2 fermi is 3.392372×10-26 kg*m3/s2. This makes H equal to 1.073105hc. Thus the coupling constant for the nuclear force is

## Conclusion

Planck's constant is a dimensioned quantity and so its magnitude can literally be any positive value. Nothing of physical significance can depend upon its magnitude. It is inversely proportional to the fine structure constant and the constant of proportionality depends upon the dimensions used.