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

Proceedings of the Royal Society of London A Mathematical and Physical Sciences

A variational method for the ground state of a Bose fluid

O. Penrose

Published May 31, 1960• 5 citations
Publisher

Abstract

Abstract A modification of the Rayleigh—Ritz variational principle is described which makes possible a calculation of the energy, wave function, and pair distribution function f12≡f(x1,x2) of a Bose fluid, such as liquid 4He, at absolute zero. The assumptions made are: (i) two-body interactions with potential Uij, (ii) trial wave functions of the form f12≡f(x1,x2) and (iii) the Kirkwood ‘superposition' approximation. Under these approximations, the expectation energy is E=12n2∫∫d3X1d3X2f12U12−h2m−1(∇18ϕ12)+(∇1ϕ12)2+n∫d3X2f13f23ϕ12⋅∇1ϕ13, where n ≡ N/V. It is shown here that making E stationary with respect to independent variations in f and ɸ corresponds to simultaneously applying the ordinary Rayleigh-Ritz principle and solving the Born-Green-Yvon integral equation for f. The method is illustrated by reproducing Bogolyubov’s results for the case where U is small. The case where U is large must be dealt with numerically, but transformations for simplifying the computations are given here.

DOI
10.1098/rspa.1960.0095

Related papers

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

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

Variational Method for Dense Systems

Vijay R. Pandharipande

The variational method for calculating energy of quantum fluids, and its applications to the Bose liquid 4He, Fermi neutron gas, and liquid 3He are discussed. The correlation functions are parameterized by their healing distance, and can depend on the states occupied by the correlated particles in the model wave function. They are calculated by constrained variation of lowest order contributions. The healing distance has a prescribed value in lowest order calculations, whereas it is sufficiently large in hopefully exact energy calculations. The direct many-body cluster diagrams are summed with successive approximations of an integral equation. The contribution of exchange diagrams is shown to decrease rapidly with the number of exchanges, and their sums are truncated after the energy has converged to within a few percent.

Symposium - International Astronomical Union•Jan 1, 1974•1 citation

Variational Study of Ground State of a Bose-Einstein Fluid

Kazuo Hiroike

Under the assumption that the ground-state wave function is expressed as a product of pair functions, the expectation value of the ground-state energy is rigorously expressed in terms of the pair distribution function. An integro-differential equation is derived for the pair distribution function from the variational principle that the expectation value of the energy is minimum. It is pointed out that the expression for the energy and the integro-differential equation become identical with those derived by Abe if the hyper-netted chain (HNC) approximation is introduced. It is proved that the HNC approximation is compatible with the virial theorem. The nature of the formulas is investigated by applying them to some cases that have been investigated by other theories. It is suggested that the present theory may in general predict the phonon-like excitations.

Progress of Theoretical Physics•Feb 1, 1962•39 citations

Theory of inhomogeneous quantum systems. I. Static properties of Bose fluids

E. Krotscheck, Guo-Xin Qian, Walter Kohn

We develop a variational theory for the determination of the ground-state properties of an inhomogeneous Bose fluid at zero temperature. Euler-Lagrange equations are derived for the one- and two-body correlations, and systematic approximation methods are physically motivated. It is shown that the optimized variational method provides a self-consistent approximate summation of ladder and ring diagrams. Numerical applications are presented for the density profile, distribution functions, energy, and chemical potential of films of $^{4}\mathrm{He}$ atoms. Both the bulk and surface energies are calculated and found to be in fair agreement with experiments.

Physical review. B, Condensed matter•Apr 1, 1985•97 citations

Variational Method for the Ground State of Multispecies Quantum Fluids

Kenneth R. Allen, Tucson Dunn

A series expansion for the ground-state energy of a multispecies quantum fluid is obtained. This expansion is derived by assuming that the ground-state wave function for the fluid can be approximated by the symmetrized ground-state wave function for a mixture of quantum ideal gases times a product of pair functions. The lowest-order approximation to the energy is minimized with respect to variations in these pair functions in order to obtain an approximation to the ground-state wave function for the system. In principle, this variational procedure can be carried out to any desired order in the energy expansion, and the machinery for doing so is explicitly exhibited. The results of this variation in the lowest order are applied to a two-species system consisting of electrons and nuclei. The results are consistent at high densities.

Physical Review•Jun 5, 1968•4 citations
Abstract A modification of the Rayleigh—Ritz variational principle is described which makes possible a calculation of the energy, wave function, and pair distribution function f12≡f(x1,x2) of a Bose fluid, such as liquid 4He, at absolute zero. The assumptions made are: (i) two-body interactions with potential Uij, (ii) trial wave functions of the form f12≡f(x1,x2) and (iii) the Kirkwood ‘superposition' approximation. Under these approximations, the expectation energy is E=12n2∫∫d3X1d3X2f12U12−h2m−1(∇18ϕ12)+(∇1ϕ12)2+n∫d3X2f13f23ϕ12⋅∇1ϕ13, where n ≡ N/V. It is shown here that making E stationary with respect to independent variations in f and ɸ corresponds to simultaneously applying the ordinary Rayleigh-Ritz principle and solving the Born-Green-Yvon integral equation for f. The method is illustrated by reproducing Bogolyubov’s results for the case where U is small. The case where U is large must be dealt with numerically, but transformations for simplifying the computations are given here.
"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."
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 variational method for calculating energy of quantum fluids, and its applications to the Bose liquid 4He, Fermi neutron gas, and liquid 3He are discussed. The correlation functions are parameterized by their healing distance, and can depend on the states occupied by the correlated particles in the model wave function. They are calculated by constrained variation of lowest order contributions. The healing distance has a prescribed value in lowest order calculations, whereas it is sufficiently large in hopefully exact energy calculations. The direct many-body cluster diagrams are summed with successive approximations of an integral equation. The contribution of exchange diagrams is shown to decrease rapidly with the number of exchanges, and their sums are truncated after the energy has converged to within a few percent.
Under the assumption that the ground-state wave function is expressed as a product of pair functions, the expectation value of the ground-state energy is rigorously expressed in terms of the pair distribution function. An integro-differential equation is derived for the pair distribution function from the variational principle that the expectation value of the energy is minimum. It is pointed out that the expression for the energy and the integro-differential equation become identical with those derived by Abe if the hyper-netted chain (HNC) approximation is introduced. It is proved that the HNC approximation is compatible with the virial theorem. The nature of the formulas is investigated by applying them to some cases that have been investigated by other theories. It is suggested that the present theory may in general predict the phonon-like excitations.
We develop a variational theory for the determination of the ground-state properties of an inhomogeneous Bose fluid at zero temperature. Euler-Lagrange equations are derived for the one- and two-body correlations, and systematic approximation methods are physically motivated. It is shown that the optimized variational method provides a self-consistent approximate summation of ladder and ring diagrams. Numerical applications are presented for the density profile, distribution functions, energy, and chemical potential of films of 4He^{4}\mathrm{He}4He atoms. Both the bulk and surface energies are calculated and found to be in fair agreement with experiments.
A series expansion for the ground-state energy of a multispecies quantum fluid is obtained. This expansion is derived by assuming that the ground-state wave function for the fluid can be approximated by the symmetrized ground-state wave function for a mixture of quantum ideal gases times a product of pair functions. The lowest-order approximation to the energy is minimized with respect to variations in these pair functions in order to obtain an approximation to the ground-state wave function for the system. In principle, this variational procedure can be carried out to any desired order in the energy expansion, and the machinery for doing so is explicitly exhibited. The results of this variation in the lowest order are applied to a two-species system consisting of electrons and nuclei. The results are consistent at high densities.