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

Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas. II. Charged Bose Gas at High Density

Élliott H. Lieb, A.Y. Sakakura

Published Feb 17, 1964• 39 citations
Publisher

Abstract

Foldy, and later Girardeau, calculated the ground-state energy of a charged Bose gas at high density. We rederive the common first term obtained by these authors by using a nonperturbation method developed previously. Our aims are: (i) to establish the validity of this common result, which has not been proved; (ii) to establish the validity and usefulness of our nonperturbation method. We also show that our method will give the correct functional dependence of the ground-state energy on the density at low density, although the exact coefficient must await a numerical computation.

DOI
10.1103/physrev.133.a899

Related papers

Ground State of the Charged Bose Gas

Marvin D. Girardeau

The ground-state energy of the charged Bose gas is calculated by the pair-correlation variational method of Girardeau and Arnowitt. The method is exact in the high-density limit (${r}_{s}\ensuremath{\ll}1$) and provides a variational extrapolation to intermediate densities. The leading terms of the high-density expansion, obtained by iteration of the variational integral equation, are ${u}_{0}=\ensuremath{-}0.804{{r}_{s}}^{\ensuremath{-}\frac{3}{4}}\ensuremath{-}(\frac{1}{8})\mathrm{ln}{r}_{s}+O(1)$, where ${u}_{0}$ is the ground-state energy per particle in Rydbergs and ${r}_{s}$ is the ratio of the mean interparticle spacing to the Bohr radius. The first term was obtained previously by Foldy, but the logarithmic term is new; it is related to screening of the long-range correlations at a distance ${r}_{0}\ensuremath{\sim}{{r}_{s}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, in analogy with the logarithmic term in the correlation energy of the electron gas. Results of numerical solutions for the intermediate-density region are presented, ranging up to ${r}_{s}=10$. On the basis of a comparison with the energy calculated from the known low-density expansion, it is estimated that the transition into Wigner's electron crystal (here a boson crystal) should occur at ${r}_{s}\ensuremath{\sim}5$.

Physical Review•Sep 1, 1962•49 citations

Ground-State Energy of the Charged Bose Gas

Donald Wright

The ground-state energy of the charged Bose gas at high densities is determined by the method of Green's functions. The pair Hamiltonian of Girardeau and Arnowitt is used to calculate the Green's functions. The leading terms in the high-density expansion are found to agree exactly with those determined by Girardeau using a variational method.

Physical Review•Mar 4, 1966•17 citations

The Ground State of the Bose Gas

Élliott H. Lieb, Robert Seiringer, Jan Philip Solovej +1 more

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, $ρ$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is $2π\hbar^2 a ρ/m$ where $a$ is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange $N^{7/5}$ law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.

arXiv:math-ph/0204027•Apr 11, 2002•5 citations

Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas

Élliott H. Lieb

The well-known first two terms in the asymptotic density series for the ground-state energy of a Bose gas, ${E}_{0}=2\ensuremath{\pi}N\ensuremath{\rho}a[1+(\frac{128}{15\ensuremath{\surd}\ensuremath{\pi}}){(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}]$, where $a$ is the scattering length of the pair potential, is ordinarily obtained by summing an infinite set of graphs in perturbation theory. We show here how this same series may be obtained by elementary methods. Our method offers the advantages of simplicity and directness. Another advantage is that the hard-core case can be handled on the same basis as a finite potential, no pseudopotential being required. In fact, the analysis of the hard-core potential turns out to be simpler than for a finite potential, as is the case in elementary quantum mechanics. In an Appendix we discuss the high-density situation and show that for a certain class of potentials Bogoliubov's theory is correct in this limit. Thus, Bogoliubov's theory, which is never correct at low density unless a pseudopotential is introduced, is really a high-density theory.

Physical Review•Jun 15, 1963•106 citations

Theory of a Charged Bose Gas. II

Chia-Wei Woo, Shang-keng Ma

The simultaneous effects of the condensate and of the long-range nature of the interaction have made the high-density charged Bose gas an interesting theoretical problem. In this paper we present a straight-forward procedure which takes these effects into account. It is based on the perturbation theory in the Bogoliubov representation and the principle of the compensation of dangerous diagrams. Both the ground-state energy and the excitation spectrum are studied. Numerical results are given for the first two orders in ${{r}_{s}}^{\frac{3}{4}}$. The ground-state energy reduces to that calculated correctly by Brueckner, and earlier by Lee and Feenberg in a variational treatment. The excitation spectrum agrees with that calculated in our preceding paper using an entirely different method.

Physical Review•Jul 5, 1967•27 citations

Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas. III. Application to the One-Dimensional Model

Élliott H. Lieb, Werner Liniger

We continue the study of the integrodifferential equation proposed previously for the evaluation of the ground-state energy of an imperfect Bose gas. We apply it here to the one-dimensional delta-function gas where the exact result is known for all values of the coupling constant $\ensuremath{\gamma}$. The results are: (i) For small $\ensuremath{\gamma}$, the equation gives the correct first two terms in an asymptotic series; (ii) a numerical solution of the equation shows that the maximum relative error occurs for $\ensuremath{\gamma}=\ensuremath{\infty}$ in which case it is 19%; (iii) for $\ensuremath{\gamma}=\ensuremath{\infty}$ we are able to compare the exact two-particle distribution function with that given by the equation. The agreement is quite good.

Physical Review•Apr 20, 1964•18 citations
Foldy, and later Girardeau, calculated the ground-state energy of a charged Bose gas at high density. We rederive the common first term obtained by these authors by using a nonperturbation method developed previously. Our aims are: (i) to establish the validity of this common result, which has not been proved; (ii) to establish the validity and usefulness of our nonperturbation method. We also show that our method will give the correct functional dependence of the ground-state energy on the density at low density, although the exact coefficient must await a numerical computation.
The ground-state energy of the charged Bose gas is calculated by the pair-correlation variational method of Girardeau and Arnowitt. The method is exact in the high-density limit (${r}_{s}\ensuremath{\ll}1$) and provides a variational extrapolation to intermediate densities. The leading terms of the high-density expansion, obtained by iteration of the variational integral equation, are ${u}_{0}=\ensuremath{-}0.804{{r}_{s}}^{\ensuremath{-}\frac{3}{4}}\ensuremath{-}(\frac{1}{8})\mathrm{ln}{r}_{s}+O(1)$, where u0{u}_{0}u0​ is the ground-state energy per particle in Rydbergs and rs{r}_{s}rs​ is the ratio of the mean interparticle spacing to the Bohr radius. The first term was obtained previously by Foldy, but the logarithmic term is new; it is related to screening of the long-range correlations at a distance ${r}_{0}\ensuremath{\sim}{{r}_{s}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, in analogy with the logarithmic term in the correlation energy of the electron gas. Results of numerical solutions for the intermediate-density region are presented, ranging up to rs=10{r}_{s}=10rs​=10. On the basis of a comparison with the energy calculated from the known low-density expansion, it is estimated that the transition into Wigner's electron crystal (here a boson crystal) should occur at ${r}_{s}\ensuremath{\sim}5$.
Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρρρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πℏ2aρ/m2π\hbar^2 a ρ/m2πℏ2aρ/m where aaa is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N7/5N^{7/5}N7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.
The well-known first two terms in the asymptotic density series for the ground-state energy of a Bose gas, ${E}_{0}=2\ensuremath{\pi}N\ensuremath{\rho}a[1+(\frac{128}{15\ensuremath{\surd}\ensuremath{\pi}}){(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}]$, where aaa is the scattering length of the pair potential, is ordinarily obtained by summing an infinite set of graphs in perturbation theory. We show here how this same series may be obtained by elementary methods. Our method offers the advantages of simplicity and directness. Another advantage is that the hard-core case can be handled on the same basis as a finite potential, no pseudopotential being required. In fact, the analysis of the hard-core potential turns out to be simpler than for a finite potential, as is the case in elementary quantum mechanics. In an Appendix we discuss the high-density situation and show that for a certain class of potentials Bogoliubov's theory is correct in this limit. Thus, Bogoliubov's theory, which is never correct at low density unless a pseudopotential is introduced, is really a high-density theory.
The simultaneous effects of the condensate and of the long-range nature of the interaction have made the high-density charged Bose gas an interesting theoretical problem. In this paper we present a straight-forward procedure which takes these effects into account. It is based on the perturbation theory in the Bogoliubov representation and the principle of the compensation of dangerous diagrams. Both the ground-state energy and the excitation spectrum are studied. Numerical results are given for the first two orders in rs34{{r}_{s}}^{\frac{3}{4}}rs​43​. The ground-state energy reduces to that calculated correctly by Brueckner, and earlier by Lee and Feenberg in a variational treatment. The excitation spectrum agrees with that calculated in our preceding paper using an entirely different method.
We continue the study of the integrodifferential equation proposed previously for the evaluation of the ground-state energy of an imperfect Bose gas. We apply it here to the one-dimensional delta-function gas where the exact result is known for all values of the coupling constant $\ensuremath{\gamma}$. The results are: (i) For small $\ensuremath{\gamma}$, the equation gives the correct first two terms in an asymptotic series; (ii) a numerical solution of the equation shows that the maximum relative error occurs for $\ensuremath{\gamma}=\ensuremath{\infty}$ in which case it is 19%; (iii) for $\ensuremath{\gamma}=\ensuremath{\infty}$ we are able to compare the exact two-particle distribution function with that given by the equation. The agreement is quite good.