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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 pθ is quantized in exactly the same manner as for the non-relativistic case; i.e.,

Δpθ = h
and hence
pθ = 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.,

K(β) = m0c²[(1−β²)−½−1]

The total energy E of the particle is then

E = K(β) + V(r)

By Bohr's analysis

ΔE = (dE/dpθ)Δpθ = h(v/r)

But (dE/dpθ) = (dE/dβ)/(dpθ/dβ)


dE/dβ = K'(β) + V'(r)(dr/dβ)
dpθ/dβ = d(mvr)/dβ = (d(mv)/dβ)r + mv(dr/dβ)


K'(β) = m0c²((−½)(−2β(1−β²)−3/2 = m0c²β(1−β²)−3/2

On the other hand,

d(mv/dβ = cd(mβ)/dβ


d(mβ)/dβ = d[m0β/(1−β²)½)]
= m0/(1−β²)½) + (−½)m0β(−β)/(1−β²)3/2)
= m0[1/(1−β²)½) + β²/(1−β²)3/2]
= m0[(1−β²+β²)/(1−β²)3/2) = m0(1−β²)−3/2)

This means that

d(mv)/dβ = m0c(1−β²)−3/2

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

dpθ/dβ = (rV'(r)/v)(dr/dβ) + rm0c(1−β²)−3/2
which, by factoring out a (r/v),
is equal to
dpθ/dβ = (r/v)[V'(r)/v)(dr/dβ) + vm0c(1−β²)−3/2]


ΔE = [V'(r)/v)(dr/dβ)
+ βm0c²(1−β²)−3/2]/{(r/v)[V'(r)/v)(dr/dβ)
+ βm0c²(1−β²)−3/2]}Δpθ
= h(v/r)

This reduces to the amazing result

Δpθ = h

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.

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