By appropriate modification of the hydrogen formula, determine the hyperfine splitting in the ground state of (a) muonic hydrogen (in which a muon—same charge and g-factor as the electron, but 207 times the mass—substitutes for the electron), (b) positronium (in which a positron—same mass and g-factor as the electron, but opposite charge—substitutes for the proton), and (c) muonium (in which an anti-muon—same mass and g-factor as a muon, but opposite charge—substitutes for the proton). Hint: Don't forget to use the reduced mass (Problem 5.1) in calculating the "Bohr radius" of these exotic "atoms," but use the actual masses in the gyromagnetic ratios. Incidentally, the answer you get for positronium (4.82×10−4 eV) is quite far from the experimental value (8.41×10−4 eV); the large discrepancy is due to pair annihilation (e++e−→γ+γ), which contributes an extra (3/4)ΔE, and does not occur (of course) in ordinary hydrogen, muonic hydrogen, or muonium.