  Combining high-temperature expansions with the recursion method and quantum
Monte Carlo simulations with the maximum entropy method, we study the dynamics
of the spin-1/2 Heisenberg chain at temperatures above and below the coupling
J. By comparing the two sets of calculations, their relative strengths are
assessed. At high temperatures, we find that there is a low-frequency peak in
the momentum integrated dynamic structure factor, due to diffusive
long-wavelength modes. This peak is rapidly suppressed as the temperature is
lowered below J. Calculation of the complete dynamic structure factor S(k,w)
shows how the spectral features associated with the two-spinon continuum
develop at low temperatures. We extract the nuclear spin-lattice relaxation
rate 1/T1 from the w-->0 limit, and compare with recent experimental results
for Sr2CuO3 and CuGeO3. We also discuss the scaling behavior of the dynamic
susceptibility, and of the static structure factor S(k) and the static
susceptibility X(k). We confirm the asymptotic low-temperature forms
S(pi)~[ln(T)]^(3/2) and X(pi)~T^(-1)[ln(T)]^(1/2), expected from previous
theoretical studies.
