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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 (meterkilogramsecond) system is 6.626×10^{−34} joulesec. 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 (centimetergramsecond) system it is 6.626×10^{−29} erg=seconds. But energy can be measured in calories, electron volts, British Thermal Units, kilowatthours 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 hbar, 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
(2.998×10^{8} 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 socalled fine structure constant, which is approximately 1/137. Its value is the ratio of the constant for the electrostatic force to the product of hbar 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 s_{0} based upon the Yukawa relation and the mass of the πmeson 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
where m_{π} is the mass of the πmesons. Note that the above formula involves hbar rather than h. Thus
It just happens that the ratio of the mass of the proton to the mass of the πmeson 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.
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 q_{1} and q_{2} separated by a distance r is given by
where (1/(4πε_{0})) is a constant equal to 9×10^{9} kg*m^{3}/s^{2}. The quantity ε_{0} 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; q_{i}=q_{e}n_{i}. Thus the force formula could be represented as
Thus the force constant in units of kg*m^{3}/s^{2} is
(1/(4πε))q_{e}²=(9×10^{9})(1.60218x10^{19})²
= 2.3103×10^{28} kg*m^{3}/s^{2}.
The ratio of this constant to hc=3.1616×10^{26} kg*m^{3}/s^{2}
is 7.34844×10^{3} or approximately 1/37.06.
Newton's Law of Gravitation is that the force between masses m_{1} and m_{2} separated by a distance r is given by
where G is a constant equal to 6.67259×10^{11} m^{3}/kg.
The mass of any body is essentially equal to the number of nucleons it contains times the mass of a nucleon; m_{i}=m_{n}n_{i}. Thus the force formula could be represented as
Thus the force constant in units of kg*m^{3}/s^{2} is Gm_{n}² = (6.67259×10^{11})(1.6749×10^{27})^{2} =1.871855×10^{64} kg*m^{3}/s^{2}.
The ratio of this constant to the product of hbar and the speed of light
in a vacuum,
hc=3.1616×10^{26} kg*m^{3}/s^{2},
is 3.76915×10^{39}. This is the coupling constant for the gravitational field.
The force between two nucleons may be given by the formula
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*m^{3}/s^{2}. This makes H equal to 1.073105hc.
Thus the coupling constant for the nuclear force is
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.
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