We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two body potential which admits a bound state. The particles are confined to a domain with Dirichlet (i.e. zero) boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross-Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain.We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two body potential which admits a bound state. The particles are confined to a domain with Dirichlet (i.e. zero) boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross-Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain.

arXiv:1608.01088Aug 3, 20160 citations

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.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.

Physical review. B, Condensed matterMay 1, 199011 citations

Elementary Aspects of the Quantum-Mechanical Many-Body Problem

Walter Greiner

Springer eBooksJan 1, 19931 citation

Elementary Aspects of the Quantum-Mechanical Many-Body Problem

Walter Greiner

Springer eBooksJan 1, 19890 citations

Subsidiary Conditions on Nuclear Many-Body Theories

J. W. Clark, MANFRED L. RISTIG

In the Jastrow theory applied to low cluster order it is necessary to restrict the variation of the two-body correlation factor $f(r)$ to the domain of functions corresponding to acceptable convergence of the cluster expansion for the energy. We examine in detail two constraints which have received some attention: the "normalization condition" and the "orthogonality (or average Pauli) conditions." The former constraint is motivated by the requirement that the liquid structure factor $S(k)$ vanish at zero wave number; the latter, by the Pauli principle in the Fermi medium. It is shown that two-body Jastrow theory cannot obey both constraints exactly for a state-independent correlation factor $f\ensuremath{\not\equiv}1$, but that the requirement $S(0)=0$ is automatically satisfied in (two-body) Brueckner reaction-matrix theory as a trivial consequence of the fulfillment, by definition of this theory, of the Pauli principle in the medium.In the Jastrow theory applied to low cluster order it is necessary to restrict the variation of the two-body correlation factor $f(r)$ to the domain of functions corresponding to acceptable convergence of the cluster expansion for the energy. We examine in detail two constraints which have received some attention: the "normalization condition" and the "orthogonality (or average Pauli) conditions." The former constraint is motivated by the requirement that the liquid structure factor $S(k)$ vanish at zero wave number; the latter, by the Pauli principle in the Fermi medium. It is shown that two-body Jastrow theory cannot obey both constraints exactly for a state-independent correlation factor $f\ensuremath{\not\equiv}1$, but that the requirement $S(0)=0$ is automatically satisfied in (two-body) Brueckner reaction-matrix theory as a trivial consequence of the fulfillment, by definition of this theory, of the Pauli principle in the medium.

Physical Review CMay 1, 197215 citations

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.

In the Jastrow theory applied to low cluster order it is necessary to restrict the variation of the two-body correlation factor

f(r)

to the domain of functions corresponding to acceptable convergence of the cluster expansion for the energy. We examine in detail two constraints which have received some attention: the "normalization condition" and the "orthogonality (or average Pauli) conditions." The former constraint is motivated by the requirement that the liquid structure factor

S(k)

vanish at zero wave number; the latter, by the Pauli principle in the Fermi medium. It is shown that two-body Jastrow theory cannot obey both constraints exactly for a state-independent correlation factor $f\ensuremath{\not\equiv}1$, but that the requirement

S(0)=0

is automatically satisfied in (two-body) Brueckner reaction-matrix theory as a trivial consequence of the fulfillment, by definition of this theory, of the Pauli principle in the medium.