New method for evaluating the density matrix and its application to the ground state form factor

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

Non-Linear Transformations and Their Applications in Many-Body Physics

Nazakat Ullah

The transformations of the type which convert an exponential into a Gaussian and vice-versa and their applications in various areas of many-body physics are discussed. A new and general method of obtaining such transformations is given using the method of moments. It is compared with other methods which could be employed to obtain such transformations. In atomic physics, we have shown how such transformations can be used to obtain electron interaction energy for the ground state of Helium and Wigner transform for the ground state of H atom. It is shown how to bring angular momentum operators to linear form so that one can use the usual property of rotation operator to calculate their matrix elements. A new way of calculating the approximate eigenvalues of a Hamiltonian is given which combines the variational principles with the principle of maximum entropy. The anharmonic oscillator Hamiltonian is used to illustrate this new method. An interesting aspect of these transformations is that one could combine them with other transformations like Grassmann integration to calculate quantities of physical interest in closed form. A general matrix element of the harmonic oscillator is given which can be used to calculate usual quantities like the trace and density matrix. Some future applications are also discussed.The transformations of the type which convert an exponential into a Gaussian and vice-versa and their applications in various areas of many-body physics are discussed. A new and general method of obtaining such transformations is given using the method of moments. It is compared with other methods which could be employed to obtain such transformations. In atomic physics, we have shown how such transformations can be used to obtain electron interaction energy for the ground state of Helium and Wigner transform for the ground state of H atom. It is shown how to bring angular momentum operators to linear form so that one can use the usual property of rotation operator to calculate their matrix elements. A new way of calculating the approximate eigenvalues of a Hamiltonian is given which combines the variational principles with the principle of maximum entropy. The anharmonic oscillator Hamiltonian is used to illustrate this new method. An interesting aspect of these transformations is that one could combine them with other transformations like Grassmann integration to calculate quantities of physical interest in closed form. A general matrix element of the harmonic oscillator is given which can be used to calculate usual quantities like the trace and density matrix. Some future applications are also discussed.

Fortschritte der Physik/Progress of PhysicsJan 1, 19941 citation

Characteristic Function Approach to Density Matrix Calculations

Metin Demіralp

Springer eBooksJan 1, 19840 citations

Some Recent Advances in Density Matrix Theory

R. McWeeny

Reviews of Modern PhysicsApr 1, 19601,460 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.