  Compared to single-component Bose-Einstein condensates, spinor Bose-Einstein
condensates display much richer dynamics. In addition to density oscillations,
spinor Bose-Einstein condensates exhibit intriguing spin dynamics that is
associated with population transfer between different hyperfine components.
This work analyzes the validity of the widely employed single-mode
approximation when describing the spin dynamics in response to a quench of the
system Hamiltonian. The single-mode approximation assumes that the different
hyperfine states all share the same time-independent spatial mode. This implies
that the resulting spin Hamiltonian only depends on the spin interaction
strength and not on the density interaction strength. Taking the spinor sodium
Bose-Einstein condensate in the $f=1$ hyperfine manifold as an example and
working within the mean-field theory framework, it is found numerically that
the single-mode approximation misses, in some parameter regimes, intricate
details of the spin and spatial dynamics. We develop a physical picture that
explains the observed phenomenon. Moreover, using that the population
oscillations described by the single-mode approximation enter into the
effective potential felt by the mean-field spinor, we derive a
semi-quantitative condition for when dynamical mean-field induced corrections
to the single-mode approximation are relevant. Our mean-field results have
implications for a variety of published and planned experimental studies.
