An ultra-cold Bose gas of two level atoms can be thought of as a spin-1/2 Bose gas. It supports spin wave collective modes due to the exchange mean field. Such collective spin oscillations have been observed in recent experiments at JILA with 87Rb atomsconfined in a harmonic trap. We present a theory of the spin wave collective modes based on the moment method for trapped gases. In the collisionless and hydrodynamic limits, we derive analytic expressions for the frequencies and damping rates of modes with dipole and quadrupole symmetry. We find that the frequency for a given mode is given by a temperature independent function of the peak density n, and falls off as 1/n. We also find that, to a very good approximation, excitations in the radial and axial directions aredecoupled. We compare our model to the numerical integration of a one dimensional version of the kinetic equation and find very goodqualitative agreement. The damping rates, however, show the largest deviation for intermediate densities, where one expects Landau damping -- which is unaccounted for in our moment approach -- to play asignificant role.
Physical Review A (Atomic, Molecular and Optical Physics)