A New Relation between Lamb Shift Energies

We derive a new relation between the observed Lamb shift energies of hydrogen and muonium atoms. The relation is based on the non-relativistic description of the Lamb shift, and the proper treatment of the reduced mass of electron and target particles (proton and muon) leads to the new formula which is expressed as $\displaystyle{{\Delta E^{(H)}_{2s_{1/2}}\over \Delta E^{(\mu)}_{2s_{1/2}}} =({1+{m_e\over m_\mu}\over 1+{m_e\over M_p}})^3}$. This relation achieves an excellent agreement with experiment and presents an important QED test free from the cutoff momentum $\Lambda$.