The Exact Number
The value of α is very close to an exact number: α = 1/137.
It was once considered possible that this exact figure could be accounted for somehow, but better theoretical and experimental research has demonstrated that the relation is inexact and that α = 1/137.035991, where again, only last digit is uncertain.
Even including all of these effects doesn’t get you everything about atoms.
Not only there is the coarse structure (from electrons orbiting nucleus), and fine structure (from relativistic effects, electron’s spin and electron’s quantum fluctuations), but there also is a hyperfine structure: interaction of electron with nuclear spin.
Spin flip transition of Hydrogen atom, for example, is the narrowest spectral line known in Physics and it’s due to this hyperfine effect that it goes even beyond the fine structure.
But the Fine Structure Constant, α, is of tremendous interest to Physics. Some have investigated whether it might not be perfectly constant.
Various measurements have indicated at various points in our scientific history that α might either vary with time or from location to location in the Universe.
Measurements of spectral lines of Hydrogen and Deuterium in some cases have indicated that perhaps α changes by ~.0001% through space or time.