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

Temperature Dependence of the Low-Momentum Excitations in a Bose Gas of Hard Spheres

Franz Mohling, Masato Morita

Published Nov 1, 1960• 19 citations
Publisher

Abstract

The pseudopotential method is used to calculate the temperature dependence of the low-momentum excitations in a Bose gas of hard spheres to an order beyond that previously known. The correction term is shown to be small compared to the leading term for all temperatures less than the critical temperature ${T}_{c}$. The excitations become poorly defined for temperatures slightly below ${T}_{c}$, however, since it is shown that in this region the excitation lifetimes are quite small.

DOI
10.1103/physrev.120.681

Related papers

Low-Lying Excitations in a Bose Gas of Hard Spheres

Franz Mohling, A. Sirlin

The pseudopotential method is used to calculate that low-lying excitation energies of a Bose gas of hard spheres at $T=0$ to an order beyond that previously calculated by this technique. The results are in agreement with those obtained by Beliaev using a different approach. The phonon velocity is found to be equal to the velocity of compressional waves to the order of approximation considered.

Physical Review•Apr 15, 1960•19 citations

Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties

T. D. Lee, Kerson Huang, Chongming Yang

It is shown that the pseudopotential method can be used for an explicit calculation of the first few terms in an expansion in power of ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$ of the eigenvalues and the corresponding eigenfunctions of a system of Bose particles with hard-sphere interaction. The low-temperature properties of the system are discussed.

Physical Review•Jun 15, 1957•1,406 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

Excitation Spectrum of an Impurity in the Hard-Sphere Bose Gas

Toshio Soda

The impurity excitation spectrum is calculated for the hard-sphere Bose gas with dilute impurity concentration, using the pseudopotential method of Lee, Huang, and Yang. The spectrum for the ideal case, in which the mass and interaction of impurity are the same with those of the bosons in the gas, is k2 in the unit of 2ma = h/ = 1 (ma being the mass of boson, and k the momentum of impurity), with an effective mass correction of −6445[(ρa3)12/π12]k2+O[(ρa3)k2,k4], where ρ is the density of the boson gas, and a is the hard-sphere diameter of the boson. The general impurity case is also calculated and found to have an impurity spectrum rk2, with a similar effective mass correction, where r is the mass ratio of the boson to the impurity.

Journal of Mathematical Physics•Jan 1, 1964•2 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

Some Nonequilibrium Properties of a Bose System of Hard Spheres at Extremely Low Temperatures

Tai Tsun Wu

The pseudopotential method is used to study a special type of flow for a Bose system of hard spheres. In the first-order approximation, the wave function of the entire system is assumed to be the product of identical single-particle wave functions, which in general are time-dependent. Such a flow is necessarily irrotational, and the single-particle wave function satisfies a Schrödinger equation with a nonlinear self-coupling term. On the basis of this equation of motion, the following properties of the Bose system are discussed: the effect of the rigid wall, the moment of inertia, the compressional wave, and a type of ``vortex filament.'' In the second-order approximation, the wave function of the system is expressed in terms of two functions such that one of them describes the single-particle state suitable for most of the particles while the other one describes the pair excitations. The much more complicated equations of motion are found, but in this approximation the flow is no longer strictly irrotational. The compressional waves are also studied in the second-order approximation.

Journal of Mathematical Physics•Jan 1, 1961•63 citations
The pseudopotential method is used to calculate the temperature dependence of the low-momentum excitations in a Bose gas of hard spheres to an order beyond that previously known. The correction term is shown to be small compared to the leading term for all temperatures less than the critical temperature Tc{T}_{c}Tc​. The excitations become poorly defined for temperatures slightly below Tc{T}_{c}Tc​, however, since it is shown that in this region the excitation lifetimes are quite small.
The pseudopotential method is used to calculate that low-lying excitation energies of a Bose gas of hard spheres at T=0T=0T=0 to an order beyond that previously calculated by this technique. The results are in agreement with those obtained by Beliaev using a different approach. The phonon velocity is found to be equal to the velocity of compressional waves to the order of approximation considered.
The impurity excitation spectrum is calculated for the hard-sphere Bose gas with dilute impurity concentration, using the pseudopotential method of Lee, Huang, and Yang. The spectrum for the ideal case, in which the mass and interaction of impurity are the same with those of the bosons in the gas, is k2 in the unit of 2ma = h/ = 1 (ma being the mass of boson, and k the momentum of impurity), with an effective mass correction of −6445[(ρa3)12/π12]k2+O[(ρa3)k2,k4], where ρ is the density of the boson gas, and a is the hard-sphere diameter of the boson. The general impurity case is also calculated and found to have an impurity spectrum rk2, with a similar effective mass correction, where r is the mass ratio of the boson to the impurity.
The pseudopotential method is used to study a special type of flow for a Bose system of hard spheres. In the first-order approximation, the wave function of the entire system is assumed to be the product of identical single-particle wave functions, which in general are time-dependent. Such a flow is necessarily irrotational, and the single-particle wave function satisfies a Schrödinger equation with a nonlinear self-coupling term. On the basis of this equation of motion, the following properties of the Bose system are discussed: the effect of the rigid wall, the moment of inertia, the compressional wave, and a type of ``vortex filament.'' In the second-order approximation, the wave function of the system is expressed in terms of two functions such that one of them describes the single-particle state suitable for most of the particles while the other one describes the pair excitations. The much more complicated equations of motion are found, but in this approximation the flow is no longer strictly irrotational. The compressional waves are also studied in the second-order approximation.