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Condition for Circular Orbit Radii in Two-Particle Systems |
Previous work showed that Bohr's initial analysis for the quantization of angular momentum
for electron orbits in atoms can be extended to any central force and to relativistic conditions.
In effect, there is a universal quantization of the angular momentum mvr for
circular orbits as hl,
where h is Planck's constant divided by 2π and l is an integer.
This leads to a quantization of orbit radii, tangential velocity, kinetic energy and potential
energy.
Let the central force field be given by a potential function V(r) where r is the distance from a particle to the center of mass of the two-particle system. The potential energy is really a function of the particle separation distance d but r is proportional to d and hence d is proportional to r. Thus in the potential energy function V*(d), d can be replaced by r to give V(r). The force on one particle is then −V'(r).
For circular orbits the centrifugal force is balanced by the attractive central force; i.e.,
This means that
Since p_{θ} is quantized to hl this means that
where c is the speed of light in a vacuum.
Allowing for the dependence of mass m on velocity the condition for the balance of centrifugal force and the attractive force
where m_{0} is the rest mass of the particle.
Let rV'(r)/(m_{0}c²) be denoted as ζ. Squaring the above equation gives
The negative solution may be ignored. The positive solution may be expressed as
There are two expression for β; i.e.,
When these are equated and the first instance of ζ replaced by its definition as rV'(r)/(m_{0}c²) the resulting equation is
This equation in turn reduces to
Since 1/ζ² equals m_{0}c²/(hcl) the above
equation, after cancellation of the term (m_{0}c²)², reduces to
This latter equation is the universal condition for the quantization of the radii of circular orbits.
However any force carried by intermediating particles will have an inverse r² dependence so the force will be of the form
where H is a constant and f(r) is normalized to f(0)=1. For this force formula the quantization condition reduces to
The expression (H/hc) is in the nature of a fine structure constant
for the force. Let (H/hc) be denoted as α and (m_{0}c²/H) as ν. Then the quantization condition
reduces to
This is the universal quantization condition for circular orbit radii.
For the electrostatic force f(r)=1 so the quantization condition reduces to
For a force carried by a decaying intermediating particle the force formula is He^{−r/r0}/r². This would include the nuclear force. For f(r)=e^{−r/r0} quantization condition reduces to the transcendental equation
For two protons subject to the attraction from their nuclear force but the repulsion from their electrostatic charge the force formula would be
(To be continued.)
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