If the angular momentum of the nucleus is I and that of the atomic electrons is J, the total angular momentum of the atom is F = I + J, and the total angular momentum quantum number f ranges from I + J to |I - J|. Show that the number of possible f values is 2I + 1 if I < J or 2J + 1 if J < I. (If you can’t find a general proof, show it for enough special cases to convince yourself of its validity.) (Because of the very small interaction of the nuclear magnetic moment with that of the electrons, a hyperfine splitting of the spectral lines is observed. When I < J, the value of I can be determined by counting the number of lines.)