The generalized momentum distribution n(p,Q), related to the half-diagonal two-body density matrix ${\mathrm{\ensuremath{\rho}}}_{2\mathrm{h}}$(
r1,
r2,${\mathbf{r}}_{1}^{\ensuremath{'}}$) by Fourier transformation in the variables
r1-${\mathbf{r}}_{1}^{\ensuremath{'}}$ and
r1-
r2, plays a key role in the description of final-state interactions in the nuclear medium and other strongly interacting many-body systems. The function n(p,Q) is explored for two Jastrow-correlated models of infinite nuclear matter within a Fermi hypernetted-chain procedure. Significant departures from ideal Fermi gas behavior in certain kinematic domains provide signatures of the strong short-range correlations contained in these models. However, such deviations are less prominent than in earlier calculations based on low-order cluster truncations; correspondingly, violations of the sequential relation are greatly reduced. Simple prescriptions for improved low-cluster-order approximations to n(p,Q) are suggested by analysis of the results of the Fermi hypernetted-chain evaluation. These results are also used to assess the quality of Silver's approximation n(p,Q)\ensuremath{\approx}n(p)[S(Q)-1] for the generalized momentum distribution in terms of the ordinary momentum distribution n(p) and the static structure function S(Q), with findings that have potential implications for the interpretation of data from inclusive electron scattering by nuclei at high momentum transfers.
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 half-diagonal two-body density matrix ${\ensuremath{\rho}}_{2h}({\mathbf{r}}_{1},{\mathbf{r}}_{2},{\mathbf{r}}_{{1}^{\ensuremath{'}}})$ is a key quantity in theoretical descriptions of nucleon knockout reactions at intermediate energies, due to its central role in treatments of the propagation and final-state interactions of ejected nucleons in the nuclear medium. This quantity is calculated for a simple model of the ground state of infinite symmetrical nuclear matter, based on a Jastrow-correlated wave function and a Fermi hypernetted-chain analysis. The dependence of ${\ensuremath{\rho}}_{2h}$ on the variables ${r}_{{11}^{\ensuremath{'}}}=|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{{1}^{\ensuremath{'}}}|,$ ${r}_{12}=|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}|,$ and the angle $\ensuremath{\theta}$ between ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{{1}^{\ensuremath{'}}}$ and ${\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}$ is investigated in some detail. Significant departures from ideal Fermi gas behavior in certain domains reflect the importance of short-range correlations and their interplay with statistical correlations. The quality of approximations to ${\ensuremath{\rho}}_{2h}$ proposed by Gersch et al., Rinat, and Silver is assessed by comparison with the Fermi hypernetted-chain results.
The microscopic study of the two-body density matrix ${\mathrm{\ensuremath{\rho}}}_{2}$(
r1,
r2,${\mathbf{r}}_{1}^{\ensuremath{'}}$,${\mathbf{r}}_{2}^{\ensuremath{'}}$) initiated for uniform Bose fluids in an earlier paper is continued for the Fermi case. We present formal results on the structure of the generalized momentum distribution n(p,q)=${\mathrm{\ensuremath{\Sigma}}}_{\mathbf{k}\mathbf{\^{}}}$〈\ensuremath{\Psi}|${\mathrm{a}}_{\mathbf{k}\mathbf{\^{}}+\mathbf{q}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$${\mathrm{a}}_{\mathbf{p}\mathbf{\^{}}\mathrm{\ensuremath{-}}\mathbf{q}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$
apˆakˆ|\ensuremath{\Psi}〉, and its Fourier inverse ${\mathrm{\ensuremath{\rho}}}_{2}$(
r1,
r2,${\mathbf{r}}_{1}^{\ensuremath{'}}$,
r2)\ensuremath{\equiv}${\mathrm{\ensuremath{\rho}}}_{2}$(
r1,
r2,${\mathbf{r}}_{1}^{\ensuremath{'}}$), based on a variational ground-state wave function of Jastrow-Slater form. The structural relations are inferred from the cluster expansions of these objects, from the asymptotic condition relating ${\mathrm{\ensuremath{\rho}}}_{2}$(
r1,
r2,${\mathbf{r}}_{1}^{\ensuremath{'}}$) to the particle density and the one-body density matrix ${\mathrm{\ensuremath{\rho}}}_{1}$(
r1,${\mathbf{r}}_{1}^{\ensuremath{'}}$), and from formal diagrammatic connections with the Bose problem. The two-body density-matrix elements ${\mathrm{\ensuremath{\rho}}}_{2}$(
r1,
r2,${\mathbf{r}}_{1}^{\ensuremath{'}}$) are thereby expressed in closed form in terms of certain sums of irreducible cluster diagrams. Some of these diagram sums are familiar from the analogous theory of the one-body density matrix; all can be evaluated quantitatively by solving a set of Fermi-hypernetted-chain (FHNC) equations. Upon invoking the sequential relation between ${\mathrm{\ensuremath{\rho}}}_{2}$(
r1,
r2,${\mathbf{r}}_{1}^{\ensuremath{'}}$) and ${\mathrm{\ensuremath{\rho}}}_{1}$(
r1,${\mathbf{r}}_{1}^{\ensuremath{'}}$), the corresponding result for the generalized momentum distribution n(p,q) effects a resolution into contributions from various scattering processes occurring in the many-body medium, specified by form factors that are susceptible to FHNC evaluation. This decomposition is comparable to that derived earlier for the Bose-fluid ground state but is complicated by contributions from exchange scattering and by a dynamically dressed Pauli kinematic correction. Silver has proposed a simple expression for the generalized momentum distribution n(p,q), a function which plays an essential role in his theory of final-state effects in deep-inelastic neutron scattering from the helium liquids. Based on the present microscopic treatment, the quality of Silver's estimate is assessed for the case of normal liquid
He3, by evaluating the necessary distribution functions and form factors within the FHNC/0 approximation.
The finite-temperature properties of Bose quantum fluids are studied by applying a variational density-matrix approach. Our ansatz for the density matrix is similar to that introduced by Campbell, K\"urten, Ristig, and Senger [Phys. Rev. 30, 3728 (1984)] but has been generalized to include three-body correlations. An expression for the finite-temperature elementary-excitation energy, \ensuremath{\epsilon}(k), is derived from an ansatz for the variational entropy. At zero temperature \ensuremath{\epsilon}(k) reduces to the same expression obtained by Brillouin-Wigner perturbation theory. The hypernetted-chain (HNC) equations provide the relations between the n-particle distribution functions and the correlation functions. A substantial set of elementary diagrams have been included to minimize the loss of precision associated with the HNC approximation. We report values for the static structure function, radial distribution function, excitation energies, particle-hole interaction, Helmholtz free energy, entropy, and isothermal sound velocity. We present a detailed study of the spectra, three-body correlations, and elementary diagram dependence of the thermodynamic spinodal curve. For relevant densities, we find that three-body correlations tend to increase the liquid-gas critical temperature
Tc while adding elementary diagrams tends to reduce
Tc. Including multiphonon terms in the spectrum produces several features in the variational entropy that were not observed at the Feynman level of excitations. Entropy isotherms show a rather abrupt transition from a low-density regime, where multiphonon contributions are insignificant, to a high-density regime where the isotherms are strongly influenced by multiphonon scattering processes.