Momentum Distribution for Bosons with Positive Scattering Length in a Trap

The coordinate-momentum double distribution function ρ(r,p)drdp is calculated in the local density approximation for bosons with positive scattering length a in a trap. The calculation is valid to the first order of a. To clarify the meaning of the result, it is compared for a special case with the double distribution function ρwd rdp of Wigner. Using the local density approximation (LDA) [1,2], which is a straightforward adaptation of the Thomas-Fermi method, the density distribution ρ(r)dr in coordinate space for BEC for a > 0 in a trap has been obtained. We want to calculate in this paper the coordinate-momentum distribution ρ(r,p)drdp in the same approximation. We follow the notation of Ref. [2] throughout. In particular, the fugacity z of the system is z = exp[μ/kT ] (1) where μ is the chemical potential. We introduce a local fugacity ζ(r) defined as ζ = z exp[−βV (r)]. (2) 1. The Gaseous Phase By the gaseous phase we include both the system before BEC sets in, and the system at high densities for the cells outside of r0 [2], i.e., outside of the region where BEC takes place. We consider such a cell of volume V in which the local fugacity is ζ . Using the method of Ref.[3] we write the grand partition function in the cell as Q = ∑ N ζTr[exp(−βH0 − βH )]. (3) The average occupation number ≪ nk ≫ of the state with momentum h̄k can be computed from Q ≪ nk ≫= ∑ N ζTr[(a†kak)exp(−βH0 − βH )]. (4) We shall drop all terms beyond the first order of H . Since H0 commutes with a † kak, we find Q = ∑ N ζTr[exp(−βH0)(1− βH )] = Q0 +Q1 (5) and Q ≪ nk ≫= ∑ N ζTr[exp(−βH0)(a†kak)(1− βH )] = A0 +A1 (6) where Q0 = ∑ N ζTr[exp(−βH0)] = ∏ [1− ζe−βε]−1 (7) 2 is the term in Q without the perturbation term. Q1 has been evaluated in Ref.[3]. Q1/Q0 = −β 4πah̄ mV 