A nonperturbative method for solving quantum field theories in three space and one time dimensions is applied to the bound-state problem of positronium and heavy quarkonia. The model includes only one dynamical photon, i.e., the irradiation channels are closed. An integral equation of the Bethe-Salpeter type is derived, being the light-cone analogue of the Tamm-Dancoff equation, and solved numerically. The model accounts for the Bohr-Sommerfeld and Dirac physics such as hyperfine splitting including the correct retardation. Special emphasis is put on the role of the Coulomb singularity in momentum space. Numerical results for the mass spectrum and the wave functions are presented, and compared to analytical results. Agreement is found for the physical value of the coupling constant. For very large coupling constants a -0.3 discrepancies are noted and discussed.

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