  The internal one-particle density matrix is discussed for Bose-Einstein
condensates with finite number of particles in a harmonic trap. The outcome of
the digonalization of the density matrix depends on the choice of the internal
coordinates: The Pethick-Pitaevskii-type internal density matrix, whose
analytical eigenvalues and eigenfunctions are evaluated, yields a fragmented
condensate, while the Jacobi-type internal density matrix leads to an ideal
condensate. We give a criterion for the choice of the internal coordinates: In
the macroscopic limit the internal density matrix should have eigenvalues and
eigenfunctions of an ideal Bose-Einstein condensate, this being a very physical
condition for cases where the system is also an ideal Bose condensation in the
laboratory frame. One choice fulfilling this boundary condition is given by the
internal Jacobi coordinates, while the internal coordinates with respect to the
center of mass do not satisfy the condition. Based on our criterion, a general
definition of the internal one-particle density matrix is presented in a
self-bound system, consisting of interacting bosons.
