Based on the relativistic Bethe-Salpeter (BS) equation, the positronium wav
efunctions in Euclidean momentum space are obtained. Meanwhile the energy l
evels of positronium ground states 1(1)S(0) and 1(3)S(1) are fitted to be 6
.7934 eV and 6.7929 eV respectively, which qualitatively agree with the pre
vious theoretical values. It is shown that the BS theory is valid and relia
ble to treat positronium.