﻿ 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 all depends upon the dimensional units. Its value is not a fundamental parameter of the universe. It is a crucial parameter for physical analysis but one that depends upon the dimensions of the system of units being used.

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

A prime candidate for such a parameter is the so-called fine structure constant, which is approximately 1/137. The electrostatic or Coulomb law for the force between two charges of magnitude q and Q is

#### F = keqQ/s²

where ke is a constant, called the Coulomb constant and s is the separation distance between the centers of the two charges. Often ke is expressed as 1/(4πε0), where ε0 is 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; i.e.,

#### e = 1.602×10−19 Coulombs

The value α of the fine structureconstant is the ratio of the constant for the electrostatic force to the product of h-bar and the speed of light c; i.e., q=ne and Q=Ne, where e is the unit of charge, the magnitude of the charge of an electron or proton. Therefore

#### F = (k0e²)nN/s²

Thus the relevant constant is (k0e²) and hence

#### α = (k0e²)/(hc)

Therefore Planck's constant should be considered as

#### h = 2π(k0e²/c)(1/α) or, equivalentlyh = (k0e²/c)(1/α)

In other words, Planck's constant is inversely proportional to the fine structure constant which is the coupling constant for the electromagnetic field. The constant of proportionality depends upon the dimensional units used.

Since Planck's constant is associated with electromagnetic fields, the question arises as to whether there are similiar constants associated with other fields such as the gravitational field or the nuclear force field.

## 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.