When solving the hydrogen atom, as a 2 body problem, we have the motion of the center of mass and the motion relative to the center of mass. The well known energy spectrum, $E_n$, that goes like $1/n^2$ is the one resulting from studying the motion relative to the center of mass.
Now the energy of the center of mass of the atom, which is purely kinetic, should be added to $E_n$ to get the full energy. So unless the atom is at rest, there will be an extra (constant) term to $E_n$
My questions are, in practice when people study the hydrogen spectrum experimentally:
1-do people look at the spectrum from 1 atom only? if this is the case, then how can they guarantee that it is not moving (to properly identify the $1/n^2$ behavior)? (very cold atom technology was not known back in the day!)
2-if they are studying a gas of hydrogen, how come the spectrum to be detected is not washed out by the transitions coming from so many atoms in all directions?
3-again for looking at hydrogen gas, how the spectrum can be studied even to see if it goes like $1/n^2$ if the atoms are moving in all directions like crazy?