  Based on our recently proposed magnon-density-waves using the microscopic
many-body approach, we investigate the longitudinal excitations in quantum
antiferromagnets by including the second order corrections in the large-$s$
expansion. The longitudinal excitation spectra for a general spin quantum
number using the antiferromagnetic Heisenberg Hamiltonian are obtained for
various spin lattice models. For bipartite lattice models, we find that the
numerical results for the energy gaps for the longitudinal modes at $q\to0$ and
the magnetic ordering wavevector $\bf Q$ are reduced by about $40-50\%$ after
including the second order corrections. Thus, our estimate of the energy gaps
for the quasi-one-dimensional (quasi-1D) antiferromagnetic compound KCuF$_3$ is
in better agreement with the experimental result. For the quasi-1D
antiferromagnets on hexagonal lattices, the full excitation spectra of both the
transverse modes (i.e., magnons) and the longitudinal modes are obtained as
functions of the nearest-neighbor coupling and the anisotropy constants. We
find two longitudinal modes due to the non-collinear nature of the triangular
antiferromagnetic order, similar to that of the phenomenological field theory
approach by Affleck. We compare our results for the longitudinal energy gaps at
the magnetic wavevectors with the experimental results for several
antiferromagnetic compounds with both integer and non-integer spin quantum
numbers, and also find good agreement after the higher-order contributions are
included in our calculations.
