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 γ F ( x , y ) of independent fermions for N particles in one dimension under harmonic confinement, the corresponding matrix γ 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 numerically. Finally, the functional relationship between γ IB ( x , y ) and γ F ( x , y ) is formally set out and illustrated.