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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 = hand 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)

Thus

Since

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

On the other hand,

Since

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

Thus

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