  In this article we introduce a multi-modal variational method to analytically
estimate the full number- and corresponding energy-density profile of a spin-1
Bose-Einstein condensate (BEC) for a number of particles as low as 500 under
harmonic confinement. To apply this method, we consider a system of spin-1 BEC
under three-dimensional isotropic and effective one-dimensional harmonic
confinement in the absence (negligible presence) of the magnetic field which
has ground state candidates of comparable energy. It should be noted that in
such circumstances kinetic energy contribution to the ground state cannot be
neglected which puts the applicability of Thomas-Fermi approximation to
question. For anti-ferromagnetic condensates, the T-F approximated energy
difference between the competing stationary states (ground state and the first
excited state) is approximately 0.3\%. As T-F approximation is only good for
condensates with a large number of particles, T-F approximated predictions can
completely go wrong especially for small condensates. This is where comes the
role of a detailed analysis using our variational method, which incorporates
the kinetic energy contribution and accurately estimates the number- and
energy-density profile even for condensates having a small number of particles.
Results of our analytical method are supported by numerical simulation. This
variational method is general and can be extended to other
similar/higher-dimensional problems to get results beyond the accuracy of the
Thomas-Fermi approximation.
