We apply the formalism of Ursell operators, introduced in a previous article, to the calculation of reduced density operators (one and two-body density operators) in a dilute quantum gas; the calculation is not a fugacity expansion and is therefore not limited to low degrees of degeneracy. We obtain quantum corrections for the one-particle density operator as a function of the second Ursell operator. For the two-body density operator, we examine how statistics and interactions combine their effects on the correlations between particles; in particular we discuss in detail how hard cores potentials affect the short range correlations, a non-perturbative effect.
An elaborate cluster analysis of the single-particle occupation probability ${n}_{\stackrel{^}{q}}$ and associated one-body density matrix
n(r) is performed for a Fermi system described by a Jastrow wave function. A diagrammatic formalism rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured, and demonstrated to convincingly high cluster order, that ${n}_{\stackrel{^}{q}}$ may be written as
n[N(q)+N1(q)], where
n is a strength factor independent of wave number
q and the quantities
N(q) and
N1(q) may be expressed as series of irreducible cluster contributions. The strength factor
n has the form
n=eQ, where
Q may also be expressed as a series of irreducible cluster contributions. Massive partial summations on the latter series yield a compact expression for
Q in terms of the spatial distribution functions corresponding to the Jastrow wave function. Working with the Fourier inverse of ${n}_{\stackrel{^}{q}}$, it is further demonstrated that
n(r) may be cast in the form $\ensuremath{\rho}n[{N}_{1}(r)+{N}_{2}(r)]\mathrm{exp}\mathcal{Q}(r)$, where $\ensuremath{\rho}$ is the particle density and the functions
N1(r),
N2(r), and
Q(r) are all given by irreducible cluster series. Massive partial summations are executed in the
Q(r) series to achieve a compact expression of this quantity in terms of the aforementioned spatial distribution functions. One has
Q(0)=Q. The leading diagrams necessary for a quantitative evaluation of the momentum distribution of liquid
3He and nuclear matter are displayed. Specialization to infinite degeneracy of the single-particle levels, while shrinking the Fermi wave number to zero (Bose limit), allows liquid
4He to be treated as well. In this limit off-diagonal long-range order appears, the condensate fraction ${\ensuremath{\rho}}^{\ensuremath{-}1}n(\ensuremath{\infty})={n}_{c}$ being just the strength factor
n. It may also be shown (under certain reasonable assumptions) that the customary ${r}^{\ensuremath{-}2}$ long-range behavior of the two-body correlations implies a singular behavior ${n}_{q}={n}_{c}(\frac{\mathrm{mc}}{2\ensuremath{\hbar}}){q}^{\ensuremath{-}1}$ of the Bose momentum distribution for small
q.
The ground-state wave function and energy of a finite system of interacting fermions are expanded in terms of multiple-particle excitations on an uncorrelated zero-order state. The resulting set of coupled equations constitutes a systematic variational generalization of Hartree-Fock theory. Comparison is made with many-body perturbation theory and it is shown that to any order the theory incorporates an infinite number of perturbation theory terms. Solutions of the equations for ground-state atomic systems are discussed and related to previous work using many-body perturbation theory. It is shown that the sums of perturbation terms necessary for convergence are automatically included in the equations for two-particle excitations. Application of the equations to open-shell atoms is described.
We present a calculation of the one-particle equal time correlation function.(x) for the one-dimensional (1D) Hubbard model in the infinite U limit. We consider the zero temperature spin incoherent regime, which is obtained by first taking the limit U --> infinity and then the limit T --> 0. Using the determinant representation for.(x), we derive analytical expressions for both large and small x at an arbitrary filling factor 0 < rho < 1/ 2. The large x asymptotics of rho(x) is found to be remarkably accurate starting from x sin(2 pi rho) similar to 1. We find that the one-particle momentum distribution function rho(k) is a smooth function of k, and. rho'(k) is peaked at k = 2k(F) in contrast to spin-coherent liquid obeying the Luttinger theorem.