The Fine-Structure Constant from the Old Quantum Theory
The Fine-Structure Constant from the Old Quantum Theory
The fine structure constant, , measures the relative strength of the electromagnetic coupling constant in quantum field theory. Its small magnitude enables very accurate predictions in the perturbation expansions of quantum electrodynamics. This famous dimensionless parameter was first introduced by Arnold Sommerfeld in 1916 in a relativistic generalization of Bohr's atomic theory. As its simplest physical realization, the fine structure constant is equal to the ratio of the speed of the electron in the first Bohr orbit to the speed of light.
α=/ℏc=0.00729735≈1/137
2
e
α
In Sommerfeld's first modification of the original atomic theory, the circular Bohr orbits were generalized so that elliptical orbits could also occur, in analogy with Kepler's laws of planetary motion. Bohr energy levels above the ground state were thereby shown to be degenerate, involving two quantum numbers, and . Sommerfeld later used the relativistic kinetic energy formula to introduce corrections to the electronic orbits. This caused some of the degenerate levels to split, thereby accounting for the "fine structure" of atomic spectral lines. Classically, the perturbation causes the relativistic elliptical orbits to precess about their major axes, although slowly compared to the electron's orbital speed.
n=1
n
ϕ
n
r
The more general significance of the fine structure constant emerged only several years after Sommerfeld introduced it. Eddington promoted the integer approximating its reciprocal (136, and later 137, as measurements became more accurate) to a near-mystical quantity, which he claimed was central to the structure of the entire universe. Pauli, for many years, sought its origin from some deeper physical principle. Today we understand that the Standard Model (SM) contains some 20 or so coupling constants, masses and mixing angles, including the fine-structure constant, which can only be experimentally determined. It is hoped that some future successor to the SM will come closer to predicting the values of these constants.
The graphics in this Demonstration show electron orbits for the principal quantum numbers , for both the nonrelativistic and relativistic theories. The quantum number determines the eccentricity via . Note that increasing corresponds to more circular orbits, in contrast to the more familiar angular-momentum quantum number , for which decreasing values give more circular orbits. The selected orbit is shown as a red curve, while the other orbits are lighter curves. For clarity, the and orbits are shown simultaneously, while the orbits are in a separate graphic. The precessional rates are exaggerated for purposes of visualization.
n=1,2,3
k=n-
n
r
ϵ=
1-/
2
k
2
n
k
l
n=1
n=2
n=3