Proton spin-lattice relaxation times have been measured in the quasi-one-dimensional
s=21 Heisenberg antiferromagnetic compounds CuSe
O4\ifmmode\cdot\else\textperiodcentered\fi{}5
H2O and CuS
O4\ifmmode\cdot\else\textperiodcentered\fi{}5
H2O, in order to study the magnetic field dependence of the 1D quantum spin dynamics at various temperatures. Information is obtained on the low-frequency behavior of the spin autocorrelation functions for the linear chains. The transverse component shows at high
T a marked decrease with magnetic field above the saturation field
Bc. Towards lower temperatures a pronounced maximum just below
Bc develops and the decrease above
Bc becomes more abrupt. The longitudinal component behaves more smoothly near and above
Bc. The field dependence of the relaxation rates is well explained at all nonzero temperatures by finite-chain calculations. For
T=0 we present analytical calculations of autocorrelation functions based on the predominance of a two-parameter spin-wave continuum for the quantum spin dynamics yielding a nondivergent behavior at $\ensuremath{\omega}=0$ and
B=0. Exact results are obtained for fields above
Bc. Further, an effective
XY approach is used for low but finite
T. Here, good agreement is found with the proton relaxation data for very low temperatures.
The behavior of a one-dimensional isotropic Heisenberg antiferromagnet of arbitrary spin in a strong magnetic field is studied in terms of an equivalent Bose-gas problem. The cases of spin 1/2 and classical spins are described by different limits of the ``Bose gas with repulsion'' model. The square-root dependence of magnetization on field is obtained in both cases, but in the quantum case one should use a properly defined effective-interaction constant, while in the classical case the result may be obtaind with a ``bare'' interaction. We find in addition that the region of linear behavior of the magnetization for classical spins is also covered partially by the Bose-gas approximation.