We show that the leading low-temperature behavior of the nth cluster coefficient ${b}_{n}$ can be calculated without sloving the $n$ -body problem, and that there is a $\mathrm{ln}T$ term in the expansion. We illustrate these results particularly for the case of ${b}_{3}$ .We show that the leading low-temperature behavior of the nth cluster coefficient ${b}_{n}$ can be calculated without sloving the $n$-body problem, and that there is a $\mathrm{ln}T$ term in the expansion. We illustrate these results particularly for the case of ${b}_{3}$.

Physical Review LettersAug 23, 197123 citations

On the cluster coefficients of a quantum gas

F. Riahi

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoMay 1, 19702 citations

Two Quantum Cluster Approximations

Thomas Maier, Orlando González, Mark Jarrell +1 more

Springer proceedings in physicsJan 1, 20034 citations

On the Quantum Theory of the Third Virial Coefficient

A. Pais, G. E. Uhlenbeck

The quantum theory of the third virial coefficient $C$ is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of ${C}_{\mathrm{BE}}$ are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for ${\mathrm{He}}^{4}$ and ${\mathrm{He}}^{3}$ , by means of a comparison of the corresponding expansion for the second virial coefficient $B$ with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to $B$ . Then the leading term of $C$ is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for $C$ is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.The quantum theory of the third virial coefficient $C$ is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of ${C}_{\mathrm{BE}}$ are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for ${\mathrm{He}}^{4}$ and ${\mathrm{He}}^{3}$, by means of a comparison of the corresponding expansion for the second virial coefficient $B$ with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to $B$. Then the leading term of $C$ is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for $C$ is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.

Physical ReviewOct 15, 195988 citations

Three-Spin System in the Presence of the Quantum Cluster Interaction

Z. Noorinezhad, B. Haghdoust, Mohammad Reza Abolhassani +2 more

Journal of Superconductivity and Novel MagnetismJun 11, 20195 citations

Quantum Corrections to the Energy Density of a Homogeneous Bose Gas

Eric Braaten, Agustín Nieto

Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of $\sqrt{ρa^3}$ , where $ρ$ is the number density and $a$ is the S-wave scattering length. We calculate the ground state energy density to second order in $\sqrt{ρa^3}$ . The coefficient of the $ρa^3$ correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on $a$ , but also on an extra parameter that describes the low energy $3\to 3$ scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is $ρa \ell_V^2$ , and $\ell_V$ is the length scale set by the van der Waals potential.Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of $\sqrt{ρa^3}$, where $ρ$ is the number density and $a$ is the S-wave scattering length. We calculate the ground state energy density to second order in $\sqrt{ρa^3}$. The coefficient of the $ρa^3$ correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on $a$, but also on an extra parameter that describes the low energy $3\to 3$ scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is $ρa \ell_V^2$, and $\ell_V$ is the length scale set by the van der Waals potential.

arXiv:cond-mat/9712041Dec 3, 19971 citation

We show that the leading low-temperature behavior of the nth cluster coefficient

{b}_{n}

can be calculated without sloving the

n

-body problem, and that there is a

\mathrm{ln}T

term in the expansion. We illustrate these results particularly for the case of

{b}_{3}

The quantum theory of the third virial coefficient

C

is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of

{C}_{\mathrm{BE}}

are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for

{\mathrm{He}}^{4}

and

{\mathrm{He}}^{3}

, by means of a comparison of the corresponding expansion for the second virial coefficient

B

with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to

B

. Then the leading term of

C

is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for

C

is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.

Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of

\sqrt{ρa^3}

, where

ρ

is the number density and

a

is the S-wave scattering length. We calculate the ground state energy density to second order in

\sqrt{ρa^3}

. The coefficient of the

ρa^3

correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on

a

, but also on an extra parameter that describes the low energy

3\to 3

scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is

ρa \ell_V^2

, and

\ell_V

is the length scale set by the van der Waals potential.