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

Abstract

To complement existing knowledge of the density matrix γF (x, y) of independent fermions for N particles in one dimension under harmonic confinement, the corresponding matrix γIB (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 numerically. Finally, the functional relationship between γIB (x, y) and γF (x, y) is formally set out and illustrated.