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Physical review. B, Solid state

Density matrix of quantum fluids

MANFRED L. RISTIG, J. W. Clark

Published Oct 1, 1976• 70 citations
Publisher

Abstract

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 $n[N(q)+{N}_{1}(q)]$, where $n$ is a strength factor independent of wave number $q$ and the quantities $N(q)$ and ${N}_{1}(q)$ may be expressed as series of irreducible cluster contributions. The strength factor $n$ has the form $n={e}^{Q}$, 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 ${N}_{1}(r)$, ${N}_{2}(r)$, and $\mathcal{Q}(r)$ are all given by irreducible cluster series. Massive partial summations are executed in the $\mathcal{Q}(r)$ series to achieve a compact expression of this quantity in terms of the aforementioned spatial distribution functions. One has $\mathcal{Q}(0)=Q$. The leading diagrams necessary for a quantitative evaluation of the momentum distribution of liquid $^{3}\mathrm{He}$ 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 $^{4}\mathrm{He}$ 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 $q$.

DOI
10.1103/physrevb.14.2875

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Pairing function for Bose fluids

MANFRED L. RISTIG

An elaborate cluster analysis of the pair occupancy ${P}_{\stackrel{^}{q}\stackrel{^}{p}}$ is performed for a quantum fluid described by a Jastrow wave function. Quantity ${P}_{\stackrel{^}{q}\stackrel{^}{p}}$ measures the simultaneous occupation of single-particle orbitals $\stackrel{^}{q}$, $\stackrel{^}{p}$ and provides information on the presence (or absence) of strongly correlated pairs in the ground state of an extended system of interacting particles. A diagrammatic formulation rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured and demonstrated to convincingly high cluster order that the pair occupancy of Bose fluids contains a finite and positive anomalous portion ${\ensuremath{\chi}}_{q}^{2}$ for pairs of particles with equal and opposite momenta, $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}+\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}=0$, in contrast to a normal Fermi fluid. The leading diagrams necessary for a structural exploration of the pairing function $\ensuremath{\chi}(r)$ which is defined as the Fourier inverse of quantity ${\ensuremath{\chi}}_{q}$ are displayed. It is further demonstrated that a close kinship exists between the function $\ensuremath{\chi}(r)$ and the one-particle density matrix $n(r)$. It is also shown that the customary ${r}^{\ensuremath{-}2}$ long-range behavior of the two-body correlation factor implies a singular behavior $\ensuremath{\sim}{q}^{\ensuremath{-}1}$ of the function ${\ensuremath{\chi}}_{q}$ for small momenta $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$. The pairing function $\ensuremath{\chi}(r)$ is calculated numerically for the ground state of liquid $^{4}\mathrm{He}$ at equilibrium density adopting the familiar short-ranged correlation factor $f(r)$ employed by Schiff and Verlet and others.

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Two-body density matrix of a normal Fermi fluid

MANFRED L. RISTIG, J. W. Clark

The microscopic study of the two-body density matrix ${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$,${\mathbf{r}}_{2}^{\ensuremath{'}}$) 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{}}}$${\mathrm{a}}_{\mathbf{p}\mathbf{\^{}}\mathrm{\ensuremath{-}}\mathbf{q}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$${\mathrm{a}}_{\mathbf{p}\mathbf{\^{}}}$${\mathrm{a}}_{\mathbf{k}\mathbf{\^{}}}$|\ensuremath{\Psi}〉, and its Fourier inverse ${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$,${\mathbf{r}}_{2}$)\ensuremath{\equiv}${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$), 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 ${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$) to the particle density and the one-body density matrix ${\mathrm{\ensuremath{\rho}}}_{1}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$), and from formal diagrammatic connections with the Bose problem. The two-body density-matrix elements ${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$) 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 ${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$) and ${\mathrm{\ensuremath{\rho}}}_{1}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$), 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 ${\mathrm{He}}^{3}$, by evaluating the necessary distribution functions and form factors within the FHNC/0 approximation.

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Physical Review•Jan 21, 1966•160 citations

Compact-Cluster Expansion for the Nuclear Many-Body Problem

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Physical Review•Dec 16, 1966•184 citations

Theory of inhomogeneous quantum systems. III. Variational wave functions for Fermi fluids

E. Krotscheck

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Physical review. B, Condensed matter•Apr 1, 1985•40 citations

Variational Theory of Quantum Fluids.

Kevin E. Schmidt

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•Aug 1, 1979•0 citations
An elaborate cluster analysis of the single-particle occupation probability {n}_{\stackrel{^}{q}} and associated one-body density matrix n(r)n(r)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)]n[N(q)+{N}_{1}(q)]n[N(q)+N1​(q)], where nnn is a strength factor independent of wave number qqq and the quantities N(q)N(q)N(q) and N1(q){N}_{1}(q)N1​(q) may be expressed as series of irreducible cluster contributions. The strength factor nnn has the form n=eQn={e}^{Q}n=eQ, where QQQ may also be expressed as a series of irreducible cluster contributions. Massive partial summations on the latter series yield a compact expression for QQQ 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)n(r)n(r) may be cast in the form \ensuremathρn[N1(r)+N2(r)]expQ(r)\ensuremath{\rho}n[{N}_{1}(r)+{N}_{2}(r)]\mathrm{exp}\mathcal{Q}(r)\ensuremathρn[N1​(r)+N2​(r)]expQ(r), where \ensuremathρ\ensuremath{\rho}\ensuremathρ is the particle density and the functions N1(r){N}_{1}(r)N1​(r), N2(r){N}_{2}(r)N2​(r), and Q(r)\mathcal{Q}(r)Q(r) are all given by irreducible cluster series. Massive partial summations are executed in the Q(r)\mathcal{Q}(r)Q(r) series to achieve a compact expression of this quantity in terms of the aforementioned spatial distribution functions. One has Q(0)=Q\mathcal{Q}(0)=QQ(0)=Q. The leading diagrams necessary for a quantitative evaluation of the momentum distribution of liquid 3He^{3}\mathrm{He}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^{4}\mathrm{He}4He to be treated as well. In this limit off-diagonal long-range order appears, the condensate fraction \ensuremathρ\ensuremath−1n(\ensuremath∞)=nc{\ensuremath{\rho}}^{\ensuremath{-}1}n(\ensuremath{\infty})={n}_{c}\ensuremathρ\ensuremath−1n(\ensuremath∞)=nc​ being just the strength factor nnn. It may also be shown (under certain reasonable assumptions) that the customary r\ensuremath−2{r}^{\ensuremath{-}2}r\ensuremath−2 long-range behavior of the two-body correlations implies a singular behavior nq=nc(mc2\ensuremathℏ)q\ensuremath−1{n}_{q}={n}_{c}(\frac{\mathrm{mc}}{2\ensuremath{\hbar}}){q}^{\ensuremath{-}1}nq​=nc​(2\ensuremathℏmc​)q\ensuremath−1 of the Bose momentum distribution for small qqq.
An elaborate cluster analysis of the pair occupancy ${P}_{\stackrel{^}{q}\stackrel{^}{p}}$ is performed for a quantum fluid described by a Jastrow wave function. Quantity ${P}_{\stackrel{^}{q}\stackrel{^}{p}}$ measures the simultaneous occupation of single-particle orbitals $\stackrel{^}{q}$, $\stackrel{^}{p}$ and provides information on the presence (or absence) of strongly correlated pairs in the ground state of an extended system of interacting particles. A diagrammatic formulation rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured and demonstrated to convincingly high cluster order that the pair occupancy of Bose fluids contains a finite and positive anomalous portion ${\ensuremath{\chi}}_{q}^{2}$ for pairs of particles with equal and opposite momenta, $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}+\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}=0$, in contrast to a normal Fermi fluid. The leading diagrams necessary for a structural exploration of the pairing function $\ensuremath{\chi}(r)$ which is defined as the Fourier inverse of quantity ${\ensuremath{\chi}}_{q}$ are displayed. It is further demonstrated that a close kinship exists between the function $\ensuremath{\chi}(r)$ and the one-particle density matrix n(r)n(r)n(r). It is also shown that the customary ${r}^{\ensuremath{-}2}$ long-range behavior of the two-body correlation factor implies a singular behavior $\ensuremath{\sim}{q}^{\ensuremath{-}1}$ of the function ${\ensuremath{\chi}}_{q}$ for small momenta $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$. The pairing function $\ensuremath{\chi}(r)$ is calculated numerically for the ground state of liquid 4He^{4}\mathrm{He}4He at equilibrium density adopting the familiar short-ranged correlation factor f(r)f(r)f(r) employed by Schiff and Verlet and others.
The microscopic study of the two-body density matrix ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}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{}}}$apˆ{\mathrm{a}}_{\mathbf{p}\mathbf{\^{}}}apˆ​akˆ{\mathrm{a}}_{\mathbf{k}\mathbf{\^{}}}akˆ​|\ensuremath{\Psi}〉, and its Fourier inverse ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$,r2{\mathbf{r}}_{2}r2​)\ensuremath{\equiv}${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}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{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$) to the particle density and the one-body density matrix ${\mathrm{\ensuremath{\rho}}}_{1}$(r1{\mathbf{r}}_{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{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}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{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$) and ${\mathrm{\ensuremath{\rho}}}_{1}$(r1{\mathbf{r}}_{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{\mathrm{He}}^{3}He3, by evaluating the necessary distribution functions and form factors within the FHNC/0 approximation.
The method of correlated basis functions provides a means of systematically exploiting our intuitive understanding of real physical systems. Here we seek especially to develop a formalism for describing systems of strongly interacting fermions, for example, nuclei and liquid He3{\mathrm{He}}^{3}He3, using a complete set of correlated functions ${\ensuremath{\Psi}}_{\mathrm{m}}=\frac{F{\ensuremath{\Phi}}_{\mathrm{m}}}{{({\ensuremath{\Phi}}_{\mathrm{m}}, {F}^{2}{\ensuremath{\Phi}}_{\mathrm{m}})}^{\frac{1}{2}}}$. FFF is taken as a product of two-body factors, $\ensuremath{\Pi}{i<j}^{}[f({r}_{\mathrm{ij}})]$, the f(rij)f({r}_{\mathrm{ij}})f(rij​) embodying at least the correlations arising from strong short-range repulsions; and ${{\ensuremath{\Phi}}_{\mathrm{m}}}$, a set of independent-particle-model functions which might serve as a basis in the absence of highly singular interactions. The matrix elements Hmn{\mathfrak{H}}_{\mathrm{mn}}Hmn​, Nmn{\mathfrak{N}}_{\mathrm{mn}}Nmn​ of the Hamiltonian and identity operators with respect to the correlated basis ${{\ensuremath{\Psi}}_{\mathrm{m}}}$ are evaluated by generalization of cluster-expansion techniques developed originally for the calculation of the partition function of a classical imperfect gas and adapted to the quantum-mechanical many-body problem by Iwamoto and Yamada. Contributions to our expansions for the Hmn{\mathfrak{H}}_{\mathrm{mn}}Hmn​ are classified according to order of magnitude in the quantity $\ensuremath{\rho}\ensuremath{\omega}$, where $\ensuremath{\omega}=\ensuremath{\int}({f}^{2}(r)\ensuremath{-}1)d\mathrm{r}$ is a "correlation parameter" and $\ensuremath{\rho}$ the average particle density. To the extent that $|\ensuremath{\rho}\ensuremath{\omega}|$ is small, it provides a true expansion parameter. A detailed technical discussion of the cluster expansions is given, with special emphasis on their symmetry properties upon truncation. Although the many-body problem is soluble by simultaneous diagonalization of the matrices (Hmn{\mathfrak{H}}_{\mathrm{mn}}Hmn​) and (Nmn{\mathfrak{N}}_{\mathrm{mn}}Nmn​), it is preferable first to transform to an orthonormal basis of "unperturbed" correlated functions ${{\ensuremath{\Theta}}_{\mathrm{m}}}$ with respect to which the Hamiltonian and identity operators have the matrix representations (Hmn{H}_{\mathrm{mn}}Hmn​) and $({N}_{\mathrm{mn}})=({\ensuremath{\delta}}_{\mathrm{mn}})$, and then to attempt the separate diagonalization of (Hmn{H}_{\mathrm{mn}}Hmn​). In general, such a transformation is expedited by the L\"owdin procedure carried out to the desired order in $\ensuremath{\rho}\ensuremath{\omega}$. However, a Schmidt orthogonalization scheme might prove more advantageous in specific cases. Field-theoretic techniques (of both perturbative and nonperturbative nature) may be brought to bear in the diagonalization of (Hmn{H}_{\mathrm{mn}}Hmn​). The inverse of the unitary transformation which takes the basis ${{\ensuremath{\Phi}}_{\mathrm{m}}}$ into the basis ${{\ensuremath{\Theta}}_{\mathrm{m}}}$, applied to the given singular Hamiltonian, leads---for any FFF in the class considered---to a non-singular "effective Hamiltonian." Following this formal cue, we set out to construct explicitly a second-quantized effective Hamiltonian which generates the same matrix elements, to prescribed order in $\ensuremath{\rho}\ensuremath{\omega}$, when operating between independent-particle kets of the model occupation-number representation, as does the given Hamiltonian in the configuration-space representation, between the corresponding correlated functions of the L\"owdin basis. The quasiparticles described by this effective Hamiltonian interact via well-behaved two-body, three-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}, QQQ-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{} potentials, in contrast to the highly singular two-body interactions of the real particles. For small $|\ensuremath{\rho}\ensuremath{\omega}|$, the effective two-body forces predominate. An avenue to deeper understanding of the applicability of the nuclear-shell model and the dressed-quasiparticle theory of liquid He3{\mathrm{He}}^{3}He3 has been opened. This article treats only the formalism for a uniform extended medium. Numerical calculations, as well as the extension of the method to finite systems, will follow.
Bethe's treatment of three-body clusters is extended to the entire linked-cluster perturbation series. The result is a completely rearranged expansion in terms of "compact clusters." The spatial correlations within these terms are all of quite short range. As a direct consequence, each of these new compact-cluster terms is proportional to ${\ensuremath{\kappa}}^{{h}^{\ensuremath{'}}\ensuremath{-}1}$, where ${h}^{\ensuremath{'}}$ is the number of momenta inside the Fermi sea which can be summed independently after allowing for momentum conservation. The "small parameter" of the expansion is $\ensuremath{\kappa}=\ensuremath{\rho}\ensuremath{\int}{|\ensuremath{\zeta}|}^{2}d\ensuremath{\tau}$, where $\ensuremath{\rho}$ is the ordinary nuclear density and $\ensuremath{\zeta}=\ensuremath{\varphi}\ensuremath{-}{\ensuremath{\psi}}_{\mathrm{BG}}$ is the "wound" in the correlated two-body (bethe-Goldstone) wave function. In nuclear matter this $\ensuremath{\kappa}$ is of order 10%. This result requires that the single-particle potentials be defined in terms of a certain subset of all the self-energy insertions. The allowed insertions are those which can be evaluated entirely on the energy shell, by means of a "generalized time-ordering" factorization. The totality of these insertions is said to constitute an "on-energy-shell mass operator" Mon{M}^{\mathrm{on}}Mon. The resulting occupied-state potentials are quantitatively very similar to those of most previous versions of nuclear-matter theory. However, the present intermediate-state potentials are small enough (being only of order +1 MeV) to be safely ignored in practical calculations. This simplification is offset by the need for a separate calculation of the three-body clusters. These single-particle potentials are "physically meaningful" in the sense that they lead to an optimum treatment of the short-range correlations. The bulk properties of nuclear systems are determined almost entirely by these short-range correlations. It is therefore proposed that similar potentials be used in a many-body theory of finite nuclei. True occupation numbers occur in a simple and natural way throughout the entire expansion. The formal and practical consequences of this feature are carefully examined. It is argued that this is significant for a theory of finite nuclei. Comparisons are made with several different "renormalized" formulations of quantum statistical mechanics, and also with the Green's-function theory of nuclear matter. The present formulation tends to artificially suppress the long-range correlations (e.g., pairing correlations) between particles near the Fermi surface. This defect can be eliminated by using a degenerate analog of the Goldstone expansion.
We develop a general variational theory for inhomogeneous Fermi systems such as the electron gas in a metal surface, the surface of liquid 3He^{3}\mathrm{He}3He, or simple models of heavy nuclei. The ground-state wave function is expressed in terms of two-body correlations, a one-body attenuation factor, and a model-system Slater determinant. Massive partial summations of cluster expansions are performed by means of Born-Green-Yvon and hypernetted-chain techniques. An optimal single-particle basis is generated by a generalized Hartree-Fock equation in which the two-body correlations screen the bare interparticle interaction. The optimization of the pair correlations leads to a state-averaged random-phase-approximation equation and a strictly microscopic determination of the particle-hole interaction.
"A theory is developed to allow the systematic calculation of the properties of quantum fluids given the microscopic two-body potential. Physical arguments are used to construct a correlation operator that approximately transforms the states of a non-interacting system into the corresponding states of the interacting-system. The theory can be regarded as a microscopic realization of Landau's assumption of a one-to-one correspondence between the non-interacting and interacting states. It is variational since the correlation operator is determined by minimizing E(p) the expectation value of the ground-state energy. We first give a survey of various other many-body theories including the simpler Jastrow variational method. Differential equations are derived to calculate the correlation operator and the related Jastrow correlation"" function. A comparison of results in several Bose systems with various methods is given. The Fermi hypernetted chain integral equations are generalized to allow the calculation of E(p) in Fermi systems, using the correlation operator wave function. The method is applied to liquid 3He and the v2-model of nuclear matter. In 3He the correlation operator gives results in much closer agreement with experiment than that with the Jastrow wave function. A new set of hypernetted chain equations are then derived to allow the calculation of the excited st;ate energies in Bose systems. The theory is then applied to the calculation of the single quasi-particle excitation spectrum of liquid 4He. The spectrum is found to be a stringent test on the form of the correlation operator, deficiencies in the correlation operator giving corresponding errors in the spectrum. Good results are obtained for the density dependence of the roton energy. A variational principle for the free energy is used to generalize the correlation operator technique to finite temperatures. Results are given for the finite temperature properties of the v8-model of nuclear matter."