I will be taking an oral exam, where I have to do some "airport physics", fast and easy magnitude estimations.
Currently I try to come up with a good way to find the Bohr radius of the hydrogen atom for this exam. Usually one would solve the differential equation by splitting the Hamiltonian in radial and angular momentum parts. But all of this needs a lot of calculations and is boring.
For the harmonic oscillator I found that one can easily recover the characteristic length by using the virial theorem and Schrödinger uncertainty relation. Is there a similarly nifty reasoning for the hydrogen atom? I think this particular method of using the virial theorem does only work for $\propto x^2$ potentials, I wasn't able to generalize it. Because of this I'm looking for a different method.
Just guessing the length by dimensional analysis does not count though, this is cheating.