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

To complement existing knowledge of the density matrix γF(x,y)γF(x,y) of independent fermions for N   particles in one dimension under harmonic confinement, the corresponding matrix γIB(x,y)γIB(x,y) for impenetrable bosons is given for N=2N=2 and 3 (with the N=4N=4 form available also). For fermions the momentum density is then obtained and illustrated numerically for N=10N=10. The boson momentum density is studied analytically at high momentum p  , the coefficients of the p−4p−4 and p−6p−6 terms being tabulated for N=2–5N=2–5 inclusive. Their dependence on powers of N   is exhibited numerically. Finally, the functional relationship between γIB(x,y)γIB(x,y) and γF(x,y)γF(x,y) is formally set out and illustrated.