  It is known that elementary bosons condense in a unique state, not so much
because this state has the lowest free particle energy but because it costs a
macroscopic amount of energy to put the particles into different states which
can then interact through quantum particle exchanges. Since individual
exchanges between the two fermions of a composite boson are ignored when
composite particles are replaced by elementary bosons, it is of importance to
reconsider the exchange energy argument for the stability of the Bose-Einstein
condensate in the case of composite bosons. We do this here in the light of the
new many-body theory which allows us to take exactly into account all possible
exchanges between the fermionic components of the composite bosons. We confirm
that the condensate of composite bosons is made of a unique state, this state
being moreover pure: a coherent superposition of states close in energy is
shown to be less favorable for both elementary and composite bosons.
