Integrity of particles, charges, angular momentum

In Newton's mechanics the idealization of a sytem moving in space and time is a 'mechanical particle' or 'mechanical point' with ideally no volume and extension of its parts and with any given mass (real number with dimension [mass]). The equations hold for any mass, including infinitesimally small masses.


Experimentally atomic and subatomic physics has taught us that: yes, there are elementary particles (leptons such as electrons, positrons, and quarks such as up-quark, down-quark, ..) which do have no extension in space⁶.


However experimentally atomic and subatomic physics has taught us the integrity of elementary particles: Never ever has a fraction of an electron been found. It was either there in the counter or not. Thus as an ingredience of the dynamic theory to be presented we cannot allow distribution functions of the mass of the particle⁷ but of distribution functions of the probability to find the integer particle somewhere in a series of experiments.


Similarly other experimentally measurable properties have been found where never ever fractions of a smallest amount have been found. Some examples are:

• Electric charge: never ever a system with less a fraction of an 'elementary charge' (that of an electron, a proton, ..) has been experimentally detected. The system can have as a charge only integer multiples of the elementary charge, including zero (no charge)⁸.
• The angular momentum of an isolated mechanical system with some extent in space (say a rotating nucleus). Never has been anything else measured as integer multiples (including zero) of the value $\hbar$.
• Same for any kind of measured action in mechanical experiments. The smallest coins are no or one $\hbar$. Thus in measuring a particle rattling in a one-dimensional box of length Lcould never be calmed down below a kinetic energy which multiplied with the orbit period $\tau$ gives $E \tau = (p^2/2m) \tau = \hbar \pi$. Thus the momentum times the path length for one orbit (moving once forth and back) is just $2 p L = 2 \pi \hbar$.

As seen from all these experiences the quantity $\hbar= 2 \pi h$, named after Max Planck is a ubiquitary 'natural constant' in mechanics. This natural constant as well as all others (including translating between different unit systems) you find at NIST (National Institute of Standards and Technology), USA). Thus we will use it often as a measuring unit, measuring thus angular momenta in multiples of $\hbar$, momenta in units of cm^-1, energy in [cm^-2 gr^-1] and so forth. This is entirely analogous to the use of the natural constant veloicity of light c= 2.999... 10¹⁰ [cm/sec] as unit for measured velocities or the gravitational constant G for using in a mass unit. Instead of the dimension units used in engineering [cgs= cm gr sec]one arrives at dimensionless quantities. Equations then become really equations between numbers. For technical applications just multiply with powers of $c, G, \hbar$ until a correct dimensional equation is reached ⁹. In addition physical relations between observables thus do not depend on the presision with which the meter in Paris or the velocity of light at Washington is measured.