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A Universal Quantization
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.,

mv²/r − V'(r) = 0
or, equivalently
mv²/r = V'(r)

This means that

mv = rV'(r)/v
and hence angular momentum
pθ=mvr is given by

pθ = r²V'(r)/v

Since pθ is quantized to hl this means that

r²V'(r)/v = hl
and therefore
v = r²V'(r)/hl
β = v/c = r²V'(r)/(hcl)

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

m0v²/(r(1−β²)½) = V'(r)
or equivalently
m0c²(v/c)²/(1−β²)½ = rV'(r)
β²/(1−β²)½ = rV'(r)/(m0c²)

where m0 is the rest mass of the particle.

Let rV'(r)/(m0c²) be denoted as ζ. Squaring the above equation gives

β4/(1−β²) = ζ²
which reduces to
the quadratic equation in β² of
β4 + ζ2β2 − ζ2 = 0
which has the solutions
β² = ½[−ζ² ± (1+4ζ²)½)]

The negative solution may be ignored. The positive solution may be expressed as

β² = ζ²[½((1+4/ζ²)½ − 1)]
or, equivalently
β = ζ[½((1+4/ζ²)½ − 1)]½

There are two expression for β; i.e.,

β = r²V'(r)/(hcl)
β = ζ[½((1+4/ζ²)½ − 1)]½

When these are equated and the first instance of ζ replaced by its definition as rV'(r)/(m0c²) the resulting equation is

rV'(r)/(m0c²)[½((1+4/ζ²)½ − 1)]½ = r²V'(r)/(hcl)
which upon cancellations and squaring produces
½((1+4/ζ²)½ − 1) = (m0c²r)²/(hcl

This equation in turn reduces to

(1+4/ζ²)½ = 1 + 2(m0c²r)²/(hcl
which upon squaring gives
1+4/ζ² = 1 + 4(m0c²r)²/(hcl)² + (m0c²r)4/(hcl)4
which upon elimination of the 1's
and cancellation of the 4's gives
1/ζ² = (m0c²r)²/(hcl)² + (m0c²r)4/(hcl)4

Since 1/ζ² equals m0c²/(hcl) the above equation, after cancellation of the term (m0c²)², reduces to

1/(rV'(r))² = r²/(hcl)² + (m0c²)²r4/(hcl)4
or, after division by r²
1/(r²V'(r))² = 1/(hcl)² + (m0c²)²r²/(hcl)4

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

V'(r) = Hf(r)/r²

where H is a constant and f(r) is normalized to f(0)=1. For this force formula the quantization condition reduces to

(1/f(r))² = (H/hc)2(1/l²) + (m0c²/H)²(H/hc)4r²/l4

The expression (H/hc) is in the nature of a fine structure constant for the force. Let (H/hc) be denoted as α and (m0c²/H) as ν. Then the quantization condition reduces to

(1/f(r))² = (α/l)2 + (α/l)4ν2r2
where l is the angular
momentum quantum number

This is the universal quantization condition for circular orbit radii.

For the electrostatic force f(r)=1 so the quantization condition reduces to

r = (1−(α/l)²)½/((α/l)²ν)

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

e2r/r0 = (α/l)2 + (α/l)4ν2r2
or, equivalently
2r/r0 = ln((α/l)²) + ln(1+(α/l)²ν²r²)

(To be continued.)

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