Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Deduction of the two-particle potential function of liquid- He4 system from scattering data

P. C. W. Fung, C. C. Lam

Published Sep 1, 1984• 3 citations
Publisher

Abstract

In treating a many-body system, the concept of correlation has been used by various authors to calculate the interacting ground state. In particular, in the Wong-Fung theory, the interacting ground state is expressed in terms of the correlation operators and the noninteracting ground state. Including pair (${u}_{2}$) and three-particle (${u}_{3}$) correlations, the wave function can be written as explicit algebraic functions of ${u}_{2}$ and ${u}_{3}$ and the noninteracting wave function [K. W. Wong et al., Phys. Rev. A. 22, 1272 (1980)]. On the other hand, the authors of the present paper have recently developed a $U$-matrix theory [Phys. Rev. A. 27, 1760 (1983)] to study general quantum-mechanical problems. For a time-independent problem, we can use this theory to find an analytic solution for the interacting wave function in terms of the potential functions of the system. A bridge between the correlation functions and the potential functions is built by identifying the interacting wave functions of the stated two theories to be the same quantity. Using the above idea, we are able to express the two-particle potential function ${v}_{2}$ as a simple function of the pair correlation ${u}_{2}$ and a "switching integral" $I$, which determines how the interaction Hamiltonian is switched on, to the lowest order to approximation. Instead of using a model for ${u}_{2}$, we obtain the explicit form of ${u}_{2}$ through the neutron scattering data of Henshaw and the concept of radial distribution function ${g}_{2}$. It is interesting to note that this ${u}_{2}$ is not divergent at any point and is Fourier transformable. Then we proceed to deduce the resulting two-particle potential function. The deduced two-particle potential is shown to be similar in form to the Lenard-Jones potential, with the same depth, a smaller hard-core radius, and a softer core.

DOI
10.1103/physreva.30.1495

Related papers

Ground State of Liquid He4

William Porter Francis, Geoffrey V. Chester, L. Reatto

A trial wave function describing the ground state of a quantum system of $N$ interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid ${\mathrm{He}}^{4}$, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.

Physical Review A•Jan 1, 1970•70 citations

A Unitary Correlation Operator Method

Hans Feldmeier, Thomas Neff, Robert Roth +2 more

The short range repulsion between nucleons is treated by a unitary correlation operator which shifts the nucleons away from each other whenever their uncorrelated positions are within the replusive core. By formulating the correlation as a transformation of the relative distance between particle pairs, general analytic expressions for the correlated wave functions and correlated operators are given. The decomposition of correlated operators into irreducible n-body operators is discussed. The one- and two-body-irreducible parts are worked out explicitly and the contribution of three-body correlations is estimated to check convergence. Ground state energies of nuclei up to mass number A=48 are calculated with a spin-isospin-dependent potential and single Slater determinants as uncorrelated states. They show that the deduced energy- and mass-number-independent correlated two-body Hamiltonian reproduces all "exact" many-body calculations surprisingly well.

•Jan 1, 1997•135 citations

Variational Theory of Quantum Fluids.

Kevin E. Schmidt

"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."

•Aug 1, 1979•0 citations

Application of Sum Rules to the Low-Temperature Interacting Boson System

Robert D. Puff

This paper is concerned with the application of sum rules satisfied by the density correlation function to the calculation of properties of the interacting Bose system, particularly liquid ${\mathrm{He}}^{4}$ and the hard-core gas. In addition to the two well-known sum rules which have been extensively discussed in the literature, we derive a third useful relation which directly involves the two-body interaction. The assumption that a single excitation exhausts all the sum rules (which is known to lead to the Feynman theory when only the first two sum rules are used) now provides, with the help of the additional sum rule, an explicit integral equation for the liquid structure factor $S(k)$. This equation can be solved quite easily for the hard-core gas, and we find the low-lying excitation spectrum to have the form ${{\ensuremath{\omega}}_{\mathrm{hc}}}^{2}(k)={k}^{2}{{v}_{T}}^{2}+{\left(\frac{{k}^{2}}{2m}\right)}^{2}+4\left(\frac{{k}^{2}}{2m}\right)\left[\left(\frac{P}{\ensuremath{\rho}}\ensuremath{-}\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)({\mathcal{P}}_{1}(\mathrm{ka})\ensuremath{-}{\mathcal{P}}_{1}(0))+\left(m{{v}_{T}}^{2}\ensuremath{-}\frac{2P}{\ensuremath{\rho}}\ensuremath{-}\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)({\mathcal{P}}_{2}(\mathrm{ka})\ensuremath{-}{\mathcal{P}}_{2}(0))\right],$ where ${v}_{T}$, $P$, $\frac{〈\mathrm{KE}〉}{〈N〉}$ and $a$ are, respectively, the sound velocity, pressure, kinetic energy per particle, and core diameter. ${\mathcal{P}}_{1}$ and ${\mathcal{P}}_{2}$ are sinusoidal functions given in the text. Although the sum rules themselves are exact, there are several difficulties associated with the use, even in the low-$k$ limit, of the sharp resonance assumption in all such integral relations. In spite of these difficulties, which are discussed briefly, we have pursued the consequences of the sharp-resonance approximation to obtain an integral equation for the liquid structure factor $S(k)$ together with the above closed-form solution for the hard-core gas excitation spectrum. The speculation that the approximation becomes essentially exact in the low-$k$ limit provides, through the third sum rule, an expression for the sound velocity in terms of the two-body potential $v(r)$ and the radial distribution function $S(r)$. We apply this relation to the case of liquid ${\mathrm{He}}^{4}$ and discuss extensions of the single-resonance approximation which can be made with the help of the third sum rule.

Physical Review•Jan 18, 1965•119 citations

Method of Correlated Basis Functions

J. W. Clark, Paul A. Westhaus

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 ${\mathrm{He}}^{3}$, using a complete set of correlated functions ${\ensuremath{\Psi}}_{\mathrm{m}}=\frac{F{\ensuremath{\Phi}}_{\mathrm{m}}}{{({\ensuremath{\Phi}}_{\mathrm{m}}, {F}^{2}{\ensuremath{\Phi}}_{\mathrm{m}})}^{\frac{1}{2}}}$. $F$ is taken as a product of two-body factors, $\ensuremath{\Pi}{i<j}^{}[f({r}_{\mathrm{ij}})]$, the $f({r}_{\mathrm{ij}})$ 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 ${\mathfrak{H}}_{\mathrm{mn}}$, ${\mathfrak{N}}_{\mathrm{mn}}$ 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 ${\mathfrak{H}}_{\mathrm{mn}}$ 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 (${\mathfrak{H}}_{\mathrm{mn}}$) and (${\mathfrak{N}}_{\mathrm{mn}}$), 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 (${H}_{\mathrm{mn}}$) and $({N}_{\mathrm{mn}})=({\ensuremath{\delta}}_{\mathrm{mn}})$, and then to attempt the separate diagonalization of (${H}_{\mathrm{mn}}$). 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 (${H}_{\mathrm{mn}}$). 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 $F$ 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{}, $Q$-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 ${\mathrm{He}}^{3}$ 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.

Physical Review•Jan 21, 1966•160 citations

SOME ASPECTS OF CORRELATION IN MANY-PARTICLE SYSTEMS.

Miller, Allen

In the formulation of the theory of the interacting many-boson system, the sign of the Fourier Transform of the two-particle interaction in the limit of long-wavelengths plays a dominant role in determining the character of the theory. With this in mind, we have obtained the following result: If the interaction between two neutral atoms is considered in the adiabatic approximation, and if the electronic wave functions correspond to the lowest energy state, then the sign of the scattering length will be negative. For the case of negative scattering lengths, an expansion is given for the ground-state energy of a system of interacting Bose particles. The expansion is performed in a way that eliminates the third-order divergence which results when the entire interaction is used as the perturbation. The structure factor, computed to first order, is in qualitative agreement with experiment, for large momentum values. For non-zero temperatures, expressions are obtained for the depletion of the zero-momentum mode and the specific heat. The results for the specific heat exhibit a discontinuity at the critical temperature. Quantitative estimates for the magnitude of the specific heat at the critical point yield results which are in fair agreement with empirical values for liquid helium. (Author)

•May 1, 1965•0 citations
In treating a many-body system, the concept of correlation has been used by various authors to calculate the interacting ground state. In particular, in the Wong-Fung theory, the interacting ground state is expressed in terms of the correlation operators and the noninteracting ground state. Including pair (u2{u}_{2}u2​) and three-particle (u3{u}_{3}u3​) correlations, the wave function can be written as explicit algebraic functions of u2{u}_{2}u2​ and u3{u}_{3}u3​ and the noninteracting wave function [K. W. Wong et al., Phys. Rev. A. 22, 1272 (1980)]. On the other hand, the authors of the present paper have recently developed a UUU-matrix theory [Phys. Rev. A. 27, 1760 (1983)] to study general quantum-mechanical problems. For a time-independent problem, we can use this theory to find an analytic solution for the interacting wave function in terms of the potential functions of the system. A bridge between the correlation functions and the potential functions is built by identifying the interacting wave functions of the stated two theories to be the same quantity. Using the above idea, we are able to express the two-particle potential function v2{v}_{2}v2​ as a simple function of the pair correlation u2{u}_{2}u2​ and a "switching integral" III, which determines how the interaction Hamiltonian is switched on, to the lowest order to approximation. Instead of using a model for u2{u}_{2}u2​, we obtain the explicit form of u2{u}_{2}u2​ through the neutron scattering data of Henshaw and the concept of radial distribution function g2{g}_{2}g2​. It is interesting to note that this u2{u}_{2}u2​ is not divergent at any point and is Fourier transformable. Then we proceed to deduce the resulting two-particle potential function. The deduced two-particle potential is shown to be similar in form to the Lenard-Jones potential, with the same depth, a smaller hard-core radius, and a softer core.
A trial wave function describing the ground state of a quantum system of NNN interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid He4{\mathrm{He}}^{4}He4, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.
The short range repulsion between nucleons is treated by a unitary correlation operator which shifts the nucleons away from each other whenever their uncorrelated positions are within the replusive core. By formulating the correlation as a transformation of the relative distance between particle pairs, general analytic expressions for the correlated wave functions and correlated operators are given. The decomposition of correlated operators into irreducible n-body operators is discussed. The one- and two-body-irreducible parts are worked out explicitly and the contribution of three-body correlations is estimated to check convergence. Ground state energies of nuclei up to mass number A=48 are calculated with a spin-isospin-dependent potential and single Slater determinants as uncorrelated states. They show that the deduced energy- and mass-number-independent correlated two-body Hamiltonian reproduces all "exact" many-body calculations surprisingly well.
"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."
This paper is concerned with the application of sum rules satisfied by the density correlation function to the calculation of properties of the interacting Bose system, particularly liquid He4{\mathrm{He}}^{4}He4 and the hard-core gas. In addition to the two well-known sum rules which have been extensively discussed in the literature, we derive a third useful relation which directly involves the two-body interaction. The assumption that a single excitation exhausts all the sum rules (which is known to lead to the Feynman theory when only the first two sum rules are used) now provides, with the help of the additional sum rule, an explicit integral equation for the liquid structure factor S(k)S(k)S(k). This equation can be solved quite easily for the hard-core gas, and we find the low-lying excitation spectrum to have the form ${{\ensuremath{\omega}}_{\mathrm{hc}}}^{2}(k)={k}^{2}{{v}_{T}}^{2}+{\left(\frac{{k}^{2}}{2m}\right)}^{2}+4\left(\frac{{k}^{2}}{2m}\right)\left[\left(\frac{P}{\ensuremath{\rho}}\ensuremath{-}\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)({\mathcal{P}}_{1}(\mathrm{ka})\ensuremath{-}{\mathcal{P}}_{1}(0))+\left(m{{v}_{T}}^{2}\ensuremath{-}\frac{2P}{\ensuremath{\rho}}\ensuremath{-}\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)({\mathcal{P}}_{2}(\mathrm{ka})\ensuremath{-}{\mathcal{P}}_{2}(0))\right],$ where vT{v}_{T}vT​, PPP, 〈KE〉〈N〉\frac{〈\mathrm{KE}〉}{〈N〉}〈N〉〈KE〉​ and aaa are, respectively, the sound velocity, pressure, kinetic energy per particle, and core diameter. P1{\mathcal{P}}_{1}P1​ and P2{\mathcal{P}}_{2}P2​ are sinusoidal functions given in the text. Although the sum rules themselves are exact, there are several difficulties associated with the use, even in the low-kkk limit, of the sharp resonance assumption in all such integral relations. In spite of these difficulties, which are discussed briefly, we have pursued the consequences of the sharp-resonance approximation to obtain an integral equation for the liquid structure factor S(k)S(k)S(k) together with the above closed-form solution for the hard-core gas excitation spectrum. The speculation that the approximation becomes essentially exact in the low-kkk limit provides, through the third sum rule, an expression for the sound velocity in terms of the two-body potential v(r)v(r)v(r) and the radial distribution function S(r)S(r)S(r). We apply this relation to the case of liquid He4{\mathrm{He}}^{4}He4 and discuss extensions of the single-resonance approximation which can be made with the help of the third sum rule.
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.
In the formulation of the theory of the interacting many-boson system, the sign of the Fourier Transform of the two-particle interaction in the limit of long-wavelengths plays a dominant role in determining the character of the theory. With this in mind, we have obtained the following result: If the interaction between two neutral atoms is considered in the adiabatic approximation, and if the electronic wave functions correspond to the lowest energy state, then the sign of the scattering length will be negative. For the case of negative scattering lengths, an expansion is given for the ground-state energy of a system of interacting Bose particles. The expansion is performed in a way that eliminates the third-order divergence which results when the entire interaction is used as the perturbation. The structure factor, computed to first order, is in qualitative agreement with experiment, for large momentum values. For non-zero temperatures, expressions are obtained for the depletion of the zero-momentum mode and the specific heat. The results for the specific heat exhibit a discontinuity at the critical temperature. Quantitative estimates for the magnitude of the specific heat at the critical point yield results which are in fair agreement with empirical values for liquid helium. (Author)