In the hydrogen atom, I believe most people are familiar with the following three equations: L^2 (psi) = l*(l+1)*hbar^2*(psi) Lz (psi) = ml*hbar*(psi) H (psi) = -1/n^2*junk*(psi) where L^2, Lz, and H are the linear operators for total angular momentum squared, angular momentum about the z-axis, and energy. I'm comfortable with eigenvalues, eigenvectors, etc. The thing I don't understand, however, is what Lz really is. Since our choice of axes is completely arbitrary, I could have just as easily chosen Lx to be Lz. But of course, if I know Lz, I can not possibly know anything about Lx (other than, perhaps, its maximum value). I gues what I'm asking is the following: is Lz just a way to say, that if we were to measure the angular momentum of an electron about a certain axis (which we'll call z) it can only have values of ml*hbar, and once we know what Lz is, we can't possibly know anything about Lx and Ly? I'd really appreciate it if somebody could straighten this out for me.