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Spin-dynamical calculations on one-dimensional systems have relied heavily on classical (s=∞) theories, despite abundant evidence that quantum effects can be extremely important at low temperatures. We present a new approach to the spin dynamics of the one-dimensional isotropic s=1/2 Heisenberg antiferromagnetic (HB AF) which does not involve the many-body techniques usually employed. It is based on analytic Bethe ansatz calculations of excitation energies and densities of states combined with finite-chain calculations of matrix elements. An important feature of our method is the use of rigorous selection rules and the introduction of new selection rules, which are valid for macroscopic systems in a magnetic field. We show that in zero field the dynamical two-spin correlation function Sμμ(q,ω) at T=0 is governed by a two-parameter continuum of spin-wave-type excitations. In nonzero field, the longitudinal component Szz(q,ω) and the transverse components Sxx(q,ω)≡Syy(q,ω) behave quite differently because they are dominated by different continua of excitations. The former is characterized by a lowest excitation branch with a zero-frequency mode moving from the zone boundary (q=π) towards the zone center (q=0) as the field increases, whereas the latter is characterized by a lowest branch with a zero frequency mode moving from q=0 to π with increasing field. The first part of our work features an approximate analytic expression for Sμμ(q,ω) at zero temperature and in zero field. Although our expression is not rigorous, exact sum rules are violated only by a small amount, and good agreement exists with the few known exact results. Our studies are extended to nonzero temperatures by placing major reliance on exact finite-chain calculations. Our work was stimulated by recent neutron scattering experiments and is oriented towards experimental comparisons. Our result for the s=1/2 integrated intensity is in much better agreement with neutron scattering data on CuCl2·2N(C5D5) (CPC) than the corresponding semiclassical result. Moreover, the spectral-weight distribution in Sμμ(q,ω) shows increasing asymmetry as q→π, a quantum effect, again in agreement with more recent neutron scattering data. The second part of our work deals with the effects of an applied magnetic field. We extend the analytic work of Ishimura and Shiba to obtain expressions for the energies and densities of states of the various excitation continua. It is shown that these continua are expected to give rise to multiple structures in the scattering intensity. Our results appear to be in quantitative agreement with preliminary results of a neutron study in CPC in a field of 70 kOe, revealing anomalous scattering intensity peaks. Our results repeatedly demonstrate the inadequacy of classical spin-wave theory for this problem. They call for additional experimental studies on quasi-one-dimensional antiferromagnets to examine other unusual and interesting phenomena predicted by our approach.