  We present an explicit and exact boost of a relativistic bound state defined
at equal time of the constituents in the Born approximation (lowest order in
hbar). To this end, we construct the Poincar\'e generators of QED and QCD in
D=1+1 dimensions, using Gauss' law to express A^0 in terms of the fermion
fields in A^1=0 gauge. We determine the fermion-antifermion bound states in the
Born approximation as eigenstates of the time and space translation generators
P^0 and P^1. The boost operator is combined with a gauge transformation so as
to maintain the gauge condition A^1=0 in the new frame. We verify that the
boosted state remains an eigenstate of P^0 and P^1 with appropriately
transformed eigenvalues and determine the transformation law of the equal-time,
relativistic wave function. The shape of the wave function is independent of
the CM momentum when expressed in terms of a variable, which is quadratically
related to the distance x between the fermions. As a consequence, the Lorentz
contraction of the wave function is proportional to 1/(E-V(x)) and thus depends
on x via the linear potential V(x).
