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

Quantum-Mechanical Many-Body Problem with Hard-Sphere Interaction

Kerson Huang, Chongming Yang

Published Feb 1, 1957• 841 citations
Publisher

Abstract

The system under consideration is an $N$-particle quantum-mechanical system enclosed in a volume $V$, in which the particles interact via two-body hard-sphere potentials, with hard-sphere diameter $a$. The two-body hard-sphere problem is first discussed by a generalization of Fermi's pseudopotential by means of which the problem is formulated entirely in terms of the scattering phase shifts. It is then shown that a pseudopotential for the $N$-body problem can be introduced, and leads to an expansion of the energy levels of the system in powers of $a$. The first order energy levels of a Bose and a Fermi system are calculated. For the Bose system, the first order energy formula exhibits an "energy gap" above the ground state, leading to properties of the system not dissimilar to that of a superfluid. The ground state energy for a Bose system is calculated to order ${a}^{3}$ and that for the Fermi system, to order ${a}^{2}$. The physical interpretation and validity of these results are discussed.

DOI
10.1103/physrev.105.767

Related papers

Many-Body Pseudopotential for Hard-Sphere Interaction

Ryuzo Abe

A many-body pseudopotential for a hard-sphere system which is mathematically equivalent to the boundary conditions imposed on the wave functions is obtained as a generalization of Huang and Yang's binary pseudopotential, and the lowest order term of the ground state energy resulting from the triple collision is calculated for the Bose system. Instead of the direct perturbational treatment of the pseudopotential, the use of the ρk-representation for the Bose system is proposed and the ground state energy is calculated in this representation. It is shown that the energy thus calculated is of the form of an expansion in powers of (ρa3)1/2, where ρ is the average density and a the hard-sphere diameter, which is agreement with Lee and Yang's conclusion drawn from the binary collision expansion method. Some discussions associated with the roton spectrum of liquid He4 are given.

Progress of Theoretical Physics•Jan 1, 1958•7 citations

Ground State of a Bose System of Hard Spheres

Tai Tsun Wu

It is shown that the pseudopotential method can be extended to yield further terms in the low-density expansion of the ground-state energy of a system of Boltzmann or Bose particles with hard-sphere interaction. Two terms beyond the known result are found, and the expansion is no longer a power series in ${({a}^{3}\ensuremath{\rho})}^{\frac{1}{2}}$. Other related properties of the system are discussed.

Physical Review•Sep 15, 1959•297 citations

Energy Levels of a Bose-Einstein System of Particles with Attractive Interactions

Kerson Huang

An $N$-body Bose-Einstein system of particles with long-range attraction and hard-sphere repulsion between particles is considered. It is shown that if the constants of the interaction have values within a certain range it is possible to calculate the ground-state energy of the system as a function of $\frac{\ensuremath{\Omega}}{N}$, where $\ensuremath{\Omega}$ is the volume of the box containing the system, in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}\ensuremath{\rightarrow}\ensuremath{\infty}$, with $\frac{N}{\ensuremath{\Omega}}=\ensuremath{\rho}$ fixed. The results show that the system can possess an $N$-body bound state, which has an equilibrium density and negative energy, and that the interactions can be saturating. Excited states are also considered. It is shown that low-lying excitations consist purely of phonons, whose velocity agrees with that computed from the macroscopic compressibility, furnished by the ground-state energy. The formula for the general excited energy levels suggests that thermodynamically the system may have a "gas" phase and two "liquid" phases, the transition between the two "liquid" phases being the analog of the Bose-Einstein condensation of the ideal gas. Thermodynamic considerations are, however, not contained in this paper.

Physical Review•Aug 15, 1959•41 citations

Quantum-Mechanical Many-Body Problem with Hard-Sphere Interaction

Ryuzo Abe

Theory of the surface charge pseudopotential for the hard-sphere potential is developed. It is shown that the formulas known to be exact are obtained by the present method. The excitation energy spectrum of the Bose system is calculated, and is shown to have the nonmonotonic behaviour similar to the roton spectrum in liquid He4. The dispersion formula for the sound velocity is calculated and compared with the derived by Landau and Khalatonikov.

Progress of Theoretical Physics•Jun 1, 1958•46 citations

Imperfect Bose Gas with Hard-Sphere Interaction

Kerson Huang, Chongming Yang, J. M. Luttinger

The method developed in the previous paper for the treatment of the quantum-mechanical $N$-body hard-sphere problem is applied to a calculation of the grand partition function of an imperfect Bose gas with hard-sphere interactions. The grand partition function is calculated to second order in an expansion in powers of $\frac{a}{\ensuremath{\lambda}}$, where $a$ is the hard-sphere diameter and $\ensuremath{\lambda}$ the thermal wavelength. The approximate equation of state for the gas phase is thereby obtained by calculating all the virial coefficients to order ${(\frac{a}{\ensuremath{\lambda}})}^{2}$.The first-order energy levels obtained in the previous paper embody some interesting physical properties. A fictitious system with exactly such energy levels is considered. The partition function for such a system can be calculated exactly and the exact equation of state obtained. It is shown that there is a phase transition, which more closely resembles an ordinary gas-liquid transition than the Bose-Einstein condensation.

Physical Review•Feb 1, 1957•186 citations

Bose System of Hard Spheres

L. Liu, Kai Wai Wong

A generalized pseudopotential for hard-sphere interaction is obtained, which is valid for all angular-momentum states. This pseudopotential is then used in the evaluation of both the ground-state energy and the excitation spectrum for a dilute Bose system of hard spheres. The calculated ground-state energy agrees with that obtained by other authors; the excitation spectrum obtained for liquid helium resembles the phonon-roton spectrum introduced by Landau.

Physical Review•Nov 1, 1963•24 citations
The system under consideration is an NNN-particle quantum-mechanical system enclosed in a volume VVV, in which the particles interact via two-body hard-sphere potentials, with hard-sphere diameter aaa. The two-body hard-sphere problem is first discussed by a generalization of Fermi's pseudopotential by means of which the problem is formulated entirely in terms of the scattering phase shifts. It is then shown that a pseudopotential for the NNN-body problem can be introduced, and leads to an expansion of the energy levels of the system in powers of aaa. The first order energy levels of a Bose and a Fermi system are calculated. For the Bose system, the first order energy formula exhibits an "energy gap" above the ground state, leading to properties of the system not dissimilar to that of a superfluid. The ground state energy for a Bose system is calculated to order a3{a}^{3}a3 and that for the Fermi system, to order a2{a}^{2}a2. The physical interpretation and validity of these results are discussed.
A many-body pseudopotential for a hard-sphere system which is mathematically equivalent to the boundary conditions imposed on the wave functions is obtained as a generalization of Huang and Yang's binary pseudopotential, and the lowest order term of the ground state energy resulting from the triple collision is calculated for the Bose system. Instead of the direct perturbational treatment of the pseudopotential, the use of the ρk-representation for the Bose system is proposed and the ground state energy is calculated in this representation. It is shown that the energy thus calculated is of the form of an expansion in powers of (ρa3)1/2, where ρ is the average density and a the hard-sphere diameter, which is agreement with Lee and Yang's conclusion drawn from the binary collision expansion method. Some discussions associated with the roton spectrum of liquid He4 are given.
An NNN-body Bose-Einstein system of particles with long-range attraction and hard-sphere repulsion between particles is considered. It is shown that if the constants of the interaction have values within a certain range it is possible to calculate the ground-state energy of the system as a function of $\frac{\ensuremath{\Omega}}{N}$, where $\ensuremath{\Omega}$ is the volume of the box containing the system, in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}\ensuremath{\rightarrow}\ensuremath{\infty}$, with $\frac{N}{\ensuremath{\Omega}}=\ensuremath{\rho}$ fixed. The results show that the system can possess an NNN-body bound state, which has an equilibrium density and negative energy, and that the interactions can be saturating. Excited states are also considered. It is shown that low-lying excitations consist purely of phonons, whose velocity agrees with that computed from the macroscopic compressibility, furnished by the ground-state energy. The formula for the general excited energy levels suggests that thermodynamically the system may have a "gas" phase and two "liquid" phases, the transition between the two "liquid" phases being the analog of the Bose-Einstein condensation of the ideal gas. Thermodynamic considerations are, however, not contained in this paper.
The method developed in the previous paper for the treatment of the quantum-mechanical NNN-body hard-sphere problem is applied to a calculation of the grand partition function of an imperfect Bose gas with hard-sphere interactions. The grand partition function is calculated to second order in an expansion in powers of $\frac{a}{\ensuremath{\lambda}}$, where aaa is the hard-sphere diameter and $\ensuremath{\lambda}$ the thermal wavelength. The approximate equation of state for the gas phase is thereby obtained by calculating all the virial coefficients to order ${(\frac{a}{\ensuremath{\lambda}})}^{2}$.The first-order energy levels obtained in the previous paper embody some interesting physical properties. A fictitious system with exactly such energy levels is considered. The partition function for such a system can be calculated exactly and the exact equation of state obtained. It is shown that there is a phase transition, which more closely resembles an ordinary gas-liquid transition than the Bose-Einstein condensation.