 Solution

The anti-muon has the same charge as a proton, so the Coulomb potential seen by the electron is the same as in the hydrogen atom. The main difference is the source mass. Separating the centre-of-mass motion replaces $m_e$ by the Coulomb two-body reduced massSince $m_{\mu }\gg m_e$, this is close to $m_e$, though less close than the electron-proton reduced mass. The smaller reduced mass gives a slightly larger Bohr radius and energies of slightly smaller magnitude. Under the assumptions in the question, the Coulomb form, angular eigenfunctions and quantum numbers are otherwise unchanged; anti-muon decay and spin effects are being neglected.