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A Generalization of Neils Bohr's Model
of Quantization for the Relativistic Case

This material generalizes the Bohr model for a particle in a central force field
when relativistic effects are taken into account. The amazing result is that angular
momentum p_{θ} is quantized in exactly the same manner as for the non-relativistic case; i.e.,

Δp_{θ} = h and hence
p_{θ} = lh

where h is Planck's constant divided by 2π and l is an
integer.

Consider a particle in a central force field with a potential energy function V(r).
The particle is in a circular orbit of radius r and has a velocity v. The relative
velocity is β=(v/c) where c is the speed of light. The kinetic energy is a
function of the relative velocity β; i.e.,

In a circular orbit mv²/r=V'(r) so mv=rV'(r)/v. Therefore

dp_{θ}/dβ = (rV'(r)/v)(dr/dβ) + rm_{0}c(1−β²)^{−3/2} which, by factoring out a (r/v), is equal to
dp_{θ}/dβ = (r/v)[V'(r)/v)(dr/dβ) + vm_{0}c(1−β²)^{−3/2}]

Thus the quantization of angular momentum in the relativistic case is independent of the potential energy function V(r)
and is exactly the same as the quantization for the non-relativistic case.