The 2pi comes from the usual method of using hbar, instead of h. Where hbar = h/2pi, and is the Planck constant adjusted to use radians instead of cycles. This

The 2pi comes from the usual method of using hbar, instead of h. Where hbar = h/2pi, and is the Planck constant adjusted to use radians instead of cycles. This is much more natural when working with frequencies instead of using cycles. Yes, this constant arises prolifically when looking at the atomic "fine structure", which modifies the usual hydrogen energies to includes interactions between an electron'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. 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/r (without all those other pesky constants embedded in Gauss's law).