The 2pi comes from the usual method of using hbar, instead of h. Where hbar = h&#x2F;2pi, and is the Planck constant adjusted to use radians instead of cycles.<p>This is much more natural when working with frequencies instead of using cycles.<p>Yes, this constant arises prolifically when looking at the atomic &quot;fine structure&quot;, which modifies the usual hydrogen energies to includes interactions between an electron&#x27;s spin and its orbit (the un-modified ones only include the kinetic energy of the reduced electron-proton and the electric potential energy). There are further interactions that can be added, eg the hyperfine interaction which includes the spin-spin interaction between the proton in the nucleus and the orbiting electron.<p>And if you use Planck units where hbar=c=G=1, you can do many things easily, for example denote the potential of an electron at distance r as just alpha&#x2F;r (without all those other pesky constants embedded in Gauss&#x27;s law).