%0 Book Section %1 statphys23_0125 %A Maghari, A. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K operator potential pressure separable statphys23 topic-8 ursell virial %T Quantum statistical mechanical properties via a separable potential of rank-two for lth partial waves %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=125 %X Nonlocal separable interactions have often been used in many-body problems because of the fact that the two-body Lippmann-Schwinger equation is exactly solvable for them. This paper describes the statistical mechanical properties for a separable potential of rank n\ for $l^th$ partial waves, which can be written as equation V = \sum_i=1^n \sum_l(2l+1)\ |\chi_il\rangle\ v_i\ łangle\chi_il\rangle\, equation can be considered as the bridge between statistical and scattering properties for the system, because some of the resolvent matrix poles are the same as the transition matrix (or scattering matrix) poles. Substituting Eq. (3) into Eq. (10) and then into Eqs. (7)-(9), the partition function, second virial coefficient and pressure virial are obtained. where $|\chi_il\rangle$ is state of the system with angular momentum quantum number $l$ and $v_i$ is the potential strength. In this work, a separable potential is used to all $l^th$ partial wave as Yamaguchi type with momentum parameters $a_1$ and $a_2$: equation \chi_il(p)\equivp|\chi_il= p^l(a_i^2+p^2)^l. equation Furthermore, the matrix elements of T\-matrix can be written as: equation p|T |p'= \sum_i=1^n \sum_l(2l+1)v_i D_ij^-1 (k^2+i\varepsilon) \chi_il^*(p)\chi_il(p') P_l(p.k), equation where equation D_il(k^2+i\varepsilon) = 1 - v_i ınt|\chi_il(q)|^2q^2 - k^2 - i\varepsilon dq. equation In thermal equilibrium,the Ursell operator $U$ , which gives the correlation between two particles, is related to the interacting and free resolvents by the Laplace transform as equation U=øint_C R(z) e^-z dz; equation equation R(z)= 1z-H - 1z-H_0, equation where the contour integral either over the complex energy parameter $z=q^2/2 \mu$. Here, the free Hamiltonian is $H_0=p^2/2 \mu$, and the interacting Hamiltonian involves the potential according to $H=H_0+V$ and $= 1/k_BT$. The system that is discussed in this paper includes pair correlations; therefore, its partition function can be written as: equation Z=Tr e^\betaH= VŁambda^3 (1+ Łambda^3V Tr U). equation where $Łambda$ is thermal deBrogile wavelength. The second virial coefficient $B(T)$ and its corresponding pressure $P_virial$, that is due to collisions, are related to the Ursell operator and the relative thermal deBrogile wavelength $Łambda_r$ by equation B(T)=-12Łambda_r^3 TrU; equation equation p_virial=n^22Łambda_r^3 Tr23H_0U equation The diagonal matrix element of resolvent in the momentum space which is given by equation łangle p|R(z)|p'\rangle=(2mp^2-q^2)^2 p|T(z)|p'\rangle equation can be considered as the bridge between statistical and scattering properties for the system, because some of the resolvent matrix poles are the same as the transition matrix (or scattering matrix) poles. Substituting (3) into (10) and then into (7)-(9), the partition function, second virial coefficient and pressure virial are obtained. References:\\ 1. A. Maghari and N. Tahmasbi, J. Phys. A, 38(2005) 4469.\\ 2. R.F. Snider, J. Chem. Phys., 88(1988) 6438.

@incollection{statphys23_0125, abstract = {Nonlocal separable interactions have often been used in many-body problems because of the fact that the two-body Lippmann-Schwinger equation is exactly solvable for them. This paper describes the statistical mechanical properties for a separable potential of rank \textit{n}\ for $l^{th}$ partial waves, which can be written as \begin{equation} \hat{V} = \sum_{i=1}^{n} \sum_{l}(2l+1)\ |\chi_{il}\rangle\ v_i\ \langle\chi_{il}\rangle\, \end{equation} can be considered as the bridge between statistical and scattering properties for the system, because some of the resolvent matrix poles are the same as the transition matrix (or scattering matrix) poles. Substituting Eq. (3) into Eq. (10) and then into Eqs. (7)-(9), the partition function, second virial coefficient and pressure virial are obtained. where $|\chi_{il}\rangle$ is state of the system with angular momentum quantum number $l$ and $v_i$ is the potential strength. In this work, a separable potential is used to all $l^{th}$ partial wave as Yamaguchi type with momentum parameters $a_1$ and $a_2$: \begin{equation} \chi_{il}(p)\equiv\langle p|\chi_{il}\rangle = \frac{p^l}{(a_i^2+p^2)^l}. \end{equation} Furthermore, the matrix elements of \textit{T}\-matrix can be written as: \begin{equation} \langle \textbf{p}|T |\textbf{p}'\rangle = \sum_{i=1}^{n} \sum_{l}(2l+1)v_i {D_{ij}^{-1} (k^2+i\varepsilon) \chi_{il}^*(p)\chi_{il}(p') \textmd{P}_l(\textbf{p}.\textbf{k}),} \end{equation} where \begin{equation} D_{il}(k^2+i\varepsilon) = 1 - v_i \int\frac{|\chi_{il}(q)|^2}{q^2 - k^2 - i\varepsilon} dq. \end{equation} In thermal equilibrium,the Ursell operator $\hat{U}$ , which gives the correlation between two particles, is related to the interacting and free resolvents by the Laplace transform as \begin{equation} \hat{U}=\oint_C \hat{R}(z) e^{-\beta z} dz; \end{equation} \begin{equation} \hat{R}(z)= \frac{1}{z-\hat{H}} - \frac{1}{z-\hat{H_0},} \end{equation} where the contour integral either over the complex energy parameter $z=q^2/2 \mu$. Here, the free Hamiltonian is $\hat{H_0}=\hat{p^2}/2 \mu$, and the interacting Hamiltonian involves the potential according to $\hat{H}=\hat{H_0}+\hat{V}$ and $\beta = 1/k_BT$. The system that is discussed in this paper includes pair correlations; therefore, its partition function can be written as: \begin{equation} Z=\textmd{Tr} e^{\beta\hat{H}}= \frac{V}{\Lambda^3} (1+ \frac{\Lambda^3}{V} \textmd{Tr} \hat{U}). \end{equation} where $\Lambda$ is thermal deBrogile wavelength. The second virial coefficient $B(T)$ and its corresponding pressure $P_{virial}$, that is due to collisions, are related to the Ursell operator and the relative thermal deBrogile wavelength $\Lambda_r$ by \begin{equation} B(T)=-\frac{1}{2}\Lambda_r^3 \textmd{Tr}\hat{U}; \end{equation} \begin{equation} p_{virial}=\frac{n^2}{2}\Lambda_r^3 \textmd{Tr}{\frac{2}{3}\hat{H_0}\hat{U}} \end{equation} The diagonal matrix element of resolvent in the momentum space which is given by \begin{equation} \langle \textbf{p}|\hat{R}(z)|\textbf{p}'\rangle=(\frac{2m}{p^2-q^2})^2 \langle \textbf{p}|\hat{T}(z)|\textbf{p}'\rangle \end{equation} can be considered as the bridge between statistical and scattering properties for the system, because some of the resolvent matrix poles are the same as the transition matrix (or scattering matrix) poles. Substituting (3) into (10) and then into (7)-(9), the partition function, second virial coefficient and pressure virial are obtained. References:\\ 1. A. Maghari and N. Tahmasbi, J. Phys. A, 38(2005) 4469.\\ 2. R.F. Snider, J. Chem. Phys., 88(1988) 6438.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Maghari, A.}, biburl = {https://www.bibsonomy.org/bibtex/2c44e36420a2f69416407a96b2ba65fe2/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {acdbdd5f2129033b6941a07be72b4c39}, intrahash = {c44e36420a2f69416407a96b2ba65fe2}, keywords = {operator potential pressure separable statphys23 topic-8 ursell virial}, month = {9-13 July}, timestamp = {2007-06-20T10:16:12.000+0200}, title = {Quantum statistical mechanical properties via a separable potential of rank-two for lth partial waves}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=125}, year = 2007 }