  We present a method for studying the excitations of low-dimensional quantum
spin systems based on the Jordan-Wigner transformation. Using an extended
RPA-scheme we calculate the correlation function of neighboring spin flips
which well approximates the optical conductivity of ${\rm Sr_2CuO_3}$. We
extend this approach to the two-leg $S=1/2$--ladder by numbering the spin
operators in a meander-like sequence. We obtain good agreement with the optical
conductivity of the spin ladder compound (La,Ca)$_{14}$Cu$_{24}$O$_{41}$ for
polarization along the rungs. For polarization along the legs higher order
correlations are important to explain the weight of high-energy continuum
excitations and we estimate the contribution of 4-- and 6--fermion processes.
