  We investigate the connection between relativistic potential models for
quark-antiquark bound states and the nonrelativistic models that have been used
successfully to fit and predict the spectra of relativistic systems. We use
Martin's operator inequality \sqrt{p^2+m^2} <= (p^2+M^2+m^2)/2M to motivate the
approximation of the relativistic kinetic energy terms in the spinless Salpeter
equation by expressions of the nonrelativistic form M+\epsilon+p^2/2M for each
quark. To investigate the validity of the resulting approximation numerically,
we generate energy spectra for q\bar{q} mesons composed of two light or two
heavy quarks using the spinless Salpeter equation with the linear-plus-Coulomb
potential, and then fit the lowest few states of each type using the effective
Schr\"{o}dinger description with the same potential. We find good fits to the
lowest four calculated c\bar{c} and the lowest three s\bar{s} states either
taking M fixed at the value M_q=\sqrt{<p^2> +m_q^2}, or allowing M_q to vary in
the fit. The energies of the lowest few c\bar{s} states are then predicted with
similar accuracy. The reasons for the success of the nonrelativistic
approximation are identified, and explain the success of Martin's
nonrelativistic predictions for the spectra of relativistic light-heavy mesons.
However, we note that the agreement between the nonrelativistic and
relativistic wave functions is not good, a point of potential concern for the
calculation of transition matrix elements.
