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Physical Review

Quantum Hard-Sphere Gas. I

Marshall Luban

Published May 17, 1965• 12 citations
Publisher

Abstract

We pose the problem of finding an operator, defined for all relative distances of two particles, which plays the role of a Hamiltonian such that the subsequent eigenvalue problem has as its only solutions precisely the eigenfunctions for two hard spheres, each of core diameter $a$. This operator is explicitly constructed. Its crucial aspect is that, despite its being defined for all relative distances ${r}_{12}$, its eigenfunctions serve as a complete set for expressing a two-particle wave function only in the restricted interval ${r}_{12}\ensuremath{\ge}a$. Subsequent papers of this series will be devoted to the application of these results to the problem of quantum hard-sphere gases.

DOI
10.1103/physrev.138.a1028

Related papers

Quantum Hard-Sphere Gas. II

Marshall Luban

Using a method due to the author which correctly reproduces the scattering of two hard spheres for all relative momenta and all partial waves, we study the properties of a low-density hard-sphere boson gas at $T=0$\ifmmode^\circ\else\textdegree\fi{}K. The system is described by a Hamiltonian $\mathcal{H}$ defined for all relative distances of the $N$ particles. The terms of $\mathcal{H}$ corresponding to the forward motion of a particle through the system lead to the notion of an effective mass ${m}^{*}$ for each particle in excess of its actual mass $m$, with $\frac{{m}^{*}}{m}$ given by a monotonically increasing function of $\ensuremath{\rho}{a}^{3}$, where $\ensuremath{\rho}$ and $a$ are the number of particles per unit volume and the hard-sphere diameter, respectively. The excitation spectrum is linear at low momenta, displays a roton-like behavior for momenta $p\ensuremath{\gtrsim}\frac{\ensuremath{\hbar}}{a}$, possesses further extrema at higher momenta, and asymptotically approaches $\frac{{p}^{2}}{2{m}^{*}}$. The pair distribution function vanishes in the low-density limit for particle separations $r\ensuremath{\le}a$.

Physical Review•May 17, 1965•6 citations

Point Transformations and the Hard-Sphere Bose Gas

F. M. Eger, Eugene P. Gross

The method of point transformations is used to provide a Hamiltonian for strongly interacting particles that is equivalent to the usual Hamiltonian. The new Hamiltonian is Hermitian, Fourier analyzable, velocity dependent, and free of the difficulties of pseudopotentials. The present work is restricted to the case that two-body encounters are dominant. The theory is expressed in the quantized wave formalism so that number-violating approximations can be applied. The well known results for the spectrum and ground-state energy of the dilute hard-sphere gas are then derived as an elementary application.

Journal of Mathematical Physics•Mar 1, 1966•17 citations

Cluster Expansion for Quantum Gases. III

Hiroaki Hara, Hiroshi Ichimura, Tohru Morita

The pair distribution function of a hard-sphere Bose gas is evaluated using a modified cluster expansion formula developed in the preceding papers of this series [H. Ichimura, Prog. Theor. Phys. 31 (1964), 538; ibid. 37 (1967), 484]. The contributions for the pair distribution function are expressed in terms of diagrams, and a partial summation of the chain-type diagrams is carried out. The temperature dependence of the resultant expression is investigated for low temperatures, at short and long distances. The result is compared with the expression for the pair distribution function which has been obtained for a weakly interacting Bose gas. The two- and three-particle distribution functions for a degenerate ideal Bose system are derived.

Progress of Theoretical Physics•Apr 1, 1968•1 citation

Perturbation theory of a nonideal Bose gas. II

Indira Ramarao, Franz Mohling

With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the radial distribution function and the structure factor through second order in the two-body interaction potential for a many-boson system below the Bose-Einstein transition temperature. For the special case of a dilute hard-sphere Bose gas at T=0° K, the structure factor is calculated in the low-momentum limit keeping terms through second order in the hard-sphere gas parameter (n a3)1/2, where n is the gas density and a is the hard-sphere diameter. The resulting expression for the structure factor is shown to satisfy the Feynman-Bijl relation explicitly.

Journal of Mathematical Physics•Dec 1, 1974•1 citation

Evaluation of the Pair Distribution Function of a Hard-Sphere Bose Gas at Zero Temperature

A. Isihara, Donald W. Jepsen

An explicit evaluation of the pair distribution function of a hard-sphere Bose gas at zero temperature is made. The result correct to first order in a hard-sphere diameter and based on a chain-diagram approximation is plotted, and the corresponding numerical values are given. Approximate analytical expressions for the distribution function at various distances are also reported.

Physical Review•Jun 5, 1967•6 citations

Many-Body Problem with Strong Forces

Robert Jastrow

A trial solution constructed from two-particle functions is applied to the problem of the $N$-particle system with strong interactions. In a variational treatment based on this wave function the expectation values, which cannot be factored into single-particle integrals, are evaluated by a cluster development in powers of the particle density. The procedure is illustrated by a calculation of the pair distribution function and zero-point energy of the hard sphere gas, for Bose and Fermi statistics.

Physical Review•Jun 1, 1955•1,148 citations
We pose the problem of finding an operator, defined for all relative distances of two particles, which plays the role of a Hamiltonian such that the subsequent eigenvalue problem has as its only solutions precisely the eigenfunctions for two hard spheres, each of core diameter aaa. This operator is explicitly constructed. Its crucial aspect is that, despite its being defined for all relative distances r12{r}_{12}r12​, its eigenfunctions serve as a complete set for expressing a two-particle wave function only in the restricted interval r12\ensuremath≥a{r}_{12}\ensuremath{\ge}ar12​\ensuremath≥a. Subsequent papers of this series will be devoted to the application of these results to the problem of quantum hard-sphere gases.
Using a method due to the author which correctly reproduces the scattering of two hard spheres for all relative momenta and all partial waves, we study the properties of a low-density hard-sphere boson gas at T=0T=0T=0\ifmmode^\circ\else\textdegree\fi{}K. The system is described by a Hamiltonian H\mathcal{H}H defined for all relative distances of the NNN particles. The terms of H\mathcal{H}H corresponding to the forward motion of a particle through the system lead to the notion of an effective mass m∗{m}^{*}m∗ for each particle in excess of its actual mass mmm, with m∗m\frac{{m}^{*}}{m}mm∗​ given by a monotonically increasing function of $\ensuremath{\rho}{a}^{3}$, where $\ensuremath{\rho}$ and aaa are the number of particles per unit volume and the hard-sphere diameter, respectively. The excitation spectrum is linear at low momenta, displays a roton-like behavior for momenta $p\ensuremath{\gtrsim}\frac{\ensuremath{\hbar}}{a}$, possesses further extrema at higher momenta, and asymptotically approaches p22m∗\frac{{p}^{2}}{2{m}^{*}}2m∗p2​. The pair distribution function vanishes in the low-density limit for particle separations $r\ensuremath{\le}a$.
The method of point transformations is used to provide a Hamiltonian for strongly interacting particles that is equivalent to the usual Hamiltonian. The new Hamiltonian is Hermitian, Fourier analyzable, velocity dependent, and free of the difficulties of pseudopotentials. The present work is restricted to the case that two-body encounters are dominant. The theory is expressed in the quantized wave formalism so that number-violating approximations can be applied. The well known results for the spectrum and ground-state energy of the dilute hard-sphere gas are then derived as an elementary application.
The pair distribution function of a hard-sphere Bose gas is evaluated using a modified cluster expansion formula developed in the preceding papers of this series [H. Ichimura, Prog. Theor. Phys. 31 (1964), 538; ibid. 37 (1967), 484]. The contributions for the pair distribution function are expressed in terms of diagrams, and a partial summation of the chain-type diagrams is carried out. The temperature dependence of the resultant expression is investigated for low temperatures, at short and long distances. The result is compared with the expression for the pair distribution function which has been obtained for a weakly interacting Bose gas. The two- and three-particle distribution functions for a degenerate ideal Bose system are derived.
With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the radial distribution function and the structure factor through second order in the two-body interaction potential for a many-boson system below the Bose-Einstein transition temperature. For the special case of a dilute hard-sphere Bose gas at T=0° K, the structure factor is calculated in the low-momentum limit keeping terms through second order in the hard-sphere gas parameter (n a3)1/2, where n is the gas density and a is the hard-sphere diameter. The resulting expression for the structure factor is shown to satisfy the Feynman-Bijl relation explicitly.
A trial solution constructed from two-particle functions is applied to the problem of the NNN-particle system with strong interactions. In a variational treatment based on this wave function the expectation values, which cannot be factored into single-particle integrals, are evaluated by a cluster development in powers of the particle density. The procedure is illustrated by a calculation of the pair distribution function and zero-point energy of the hard sphere gas, for Bose and Fermi statistics.