  We present an investigation of various gauge invariant definitions of the
$q\bar q$ Bethe-Salpeter (BS) amplitude for mesons in lattice QCD, and compare
them to the Coulomb and Landau gauge BS amplitudes. We show that the gauge
invariant BS amplitude is considerably broadened by the use of ``fat'' gauge
links (constructed by smearing the links of the original lattice). A
qualitative demonstration of the Lorentz contraction of the pion wavefunction
at non-zero momentum is given. We also calculate density-density correlations
and discuss the limitations in extracting the charge radius of the pion from
these observables. Lastly, the polarization dependence of the BS amplitude for
the $\rho$ meson is exhibited, and we extract the relative admixture of $l=0$
and $l=2$ states showing that simple hydrogen-like wavefunctions provide a good
fit.
