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A Correspondence Between Statically and Dynamically Determined Fields Due to Particles |
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Consider the following two situations in mathematical physics.
Suppose the function giving the physical dF on a unit charge at position Z due to a charge of dq located at position ζ is given by
where s is the distance between point Z and point ζ. This distance is given by |Z−ζ|. The increment of charge is given by ρ(ζ)dR.
The intensity vector G of the field at Z is then given by
For the static situation the intensity at Z is constant so the expected value of G, E{ }, is just G; i.e.,
On the other hand in the dynamic situation the field intensity at Z fluctuates. When the charge Q is at point ζ the force on a unit charge at point Z is given by
Let P(ζ) be the probability density function representing the proportion of the time the charge spends at point ζ. For a particle traveling along a path at a velocity of v the probability density function is given by
where T is the total time required to traverse the path.
The expected value of the vector of field intensity H is then given by
The expected value in this case is just the same as the time-averaged value.
The rate of rotation of an electron in a hydrogen atom, according to the Bohr model, is easily determined. The electrostatic force between an electron and a proton has to be balanced by the centrifugal force on the electron; i.e.,
where G is a constant, e is the electrical charge of the electron and the proton, m is the mass of the electron and r is the orbit radius for the electron. Therefore
The frequency ν is equal to ω/(2π). This works out to be 6.6×10^{15} times per second; i.e., 6.6 quadrillion times per second. Any observation will involve a time-average and at the above frequency the observation will be equal to the expected value. Thus the observed world is the world of the dynamic appearances of physical systems. In the case of electrons this would tubular elliptical rings.
At the nuclear level Åge Bohr and Ben Motellsen found that rotations satisfied the equation
where E_{rot} is the energy of rotation, h is Planck's constant divided by 2π, J is the moment of inertia
and I is an integer. This is equivalent to the angular of momentum L of the system being equal to
Since L is equal to Jω
The smaller the scale of a system the smaller is its moment of inertia and thus the more rapid is its rate of rotation. The moment of inertia of a system is directly porportional to the mass of the system and to the square of its scale. The ratio of the radii of an atom to that of its nucleus is on the order of 10^{5} where as the ratio of the mass of the outer band of electrons of an atom to that of the nucleus is on the order of 10^{−3}. Thus the moment of inertia of the electrons in an atom is about 10^{7} times greater than the moment of inertia of the nucleus. Thus nuclear particles are involved in rotations at rates about ten million times higher than those of the electrons. Therefore the time-averaged observations for atomic and subatomic systems are essentially the same as their static equivalents. In other words, it is impossible to distinguish between a particle engaged in periodic motion and that the static equivalent. When quantum analysis relies upon the time-independent Schrödinger equation the solution corresponds to the static equivalent of the dynamic system. It gets the blurred disk of the rapidly rotating fan and not the rotating fan itself.
Note that if the charge density ρ(ζ) is given by P(ζ)Q then the dynamic case problem is mathematically identical to the static case problem. This correspondence points up the significance of the time-spent probability density function to mechanics. It is as significant as the spatial distribution of any charge.
From this correspondence it then follows that if the force law is strictly an inverse distance square law then the effect of a particle that uniformly traverses a spherical surface is the same, on average outside of the sphere as if it were located at the center of the sphere. Inside the sphere the effect is zero. This is relevant for the field due to the positive electrical charge of the protons in nuclei.
The so-called strong nuclear force is not a strictly inverse distance squared force because the particles carrying the force, the pi-mesons, decay over time and hence with distance. This makes the likely functional form inverse distance squared weighted by a negative exponential; i.e. H·exp(−s/s_{0})/s², where H and s_{0} are parameters.
The above derivation which was for a single particle applies equally as well to multiple particles. The multiple particles may be independent or interlinked. As a particular case it would apply to structures that rotate.
Thus for n particles with spent-time probability density functions and charges of P_{I} and Q_{I}, respectively. for i=1...n the equivalent charge density ρ(ζ) is given by
It is not easy to find a static structure for charges that is spherical. It is however easy to specify a dynamic structure that produces the appearance of a sphere. For example, a circular ring that is flipping rapidly over and over like a flipped coin gives that spherical appearance. If the ring is a ring of linked particles and it is rotating about an axis through its center perpendicular to its plane then the particles appear at all different points of the sphere of rotation.
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