First-order density matrices in one dimension for independent fermions and impenetrable bosons in harmonic traps

To complement existing knowledge of the density matrix gamma(F)(x, y) of independent fermions for N particles in one dimension under harmonic confinement, the corresponding matrix gamma(IB)(x, y) for impenetrable bosons is given for N = 2 and 3 (with the N = 4 form available also). For fermions the momentum density is then obtained and illustrated numerically for N = 10. The boson momentum density is studied analytically at high momentum p; the coefficients of the p(-4) and p(-6) terms being tabulated for N = 2-5 inclusive. Their dependence on powers of N is exhibited numericallyi Finally, the functional relationship between M(x. y) and yF(x, y) is formally set out and illustrated. (c) 2006 Elsevier B.V. All rights reserved.