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Scattering on the resonance state of a Bose gas

To describe the lowest-lying gas-like state of the Hamiltonian (4.18), we use for the one-body Bijl-Jastrow term an ansatz similar to (2.41):

$\displaystyle f_1(\r) = \exp\left\{-\frac{x^2+y^2}{2\alpha_\rho^2}-\frac{z^2}{2\alpha_z^2}\right\}$     (2.89)

Here, $\alpha _z$ and $\alpha_\rho$ determine the Gaussian width of $\psi _T$ in the longitudinal and transverse direction, respectively. These variational parameters $\alpha _z$ and $\alpha_{\rho}$ are optimized in the course of the VMC calculation by minimizing the energy expectation value. The two-body correlation factor $f_2(r)$ (2.37) is chosen to reproduce closely the scattering behavior of two bosons at low energies. For the hard-sphere potential (1.48), we take
\begin{displaymath}
f_2(\r)=
\left\{
\begin{array}{cl}
0,& \vert\r\vert \le a_{3...
..._{3D}}/{\vert\r\vert},& \vert\r\vert>a_{3D}
\end{array}\right.
\end{displaymath} (2.90)

The constraint $f_2=0$ for $r \le a_{3D}$ accounts for the boundary condition imposed by the hard-sphere potential, it is exact even for the many-body system. For the short-range potential (1.97), we use instead

\begin{displaymath}
f_2(\r)=
\left\{
\begin{array}{cl}
0,& \frac{x^2 + y^2}{a^2}...
...},& \frac{x^2 + y^2}{a^2}+\frac{z^2}{b^2}>1
\end{array}\right.
\end{displaymath} (2.91)

where $a$ and $b$ denote the lengths of the semi-axes of an ellipse. For two particles under highly-elongated confinement, the nodal surface is to a good approximation ellipticly shaped as will be discussed in Sec. 4.4.1. Thus, the parameters $a$ and $b$ are determined by fitting the elliptical surface to the nodal surface obtained by solving the Schrödinger equation for $N=2$, Eqs. 4.7 and 4.8, by performing a B-spline basis set calculation. In contrast to $V^{HS}$, the constraint $f_2=0$ in Eq. 2.91 parameterizes the many-body nodal surface for $V^{SR}$ only approximately. We expect that our parameterization leads to an accurate description of quasi-1D Bose gases if the average distance between particles is much larger than the semi-axes of the ellipse. The trial wave functions discussed here in the context of our VMC calculations also enter our FN-DMC calculations.


next up previous contents
Next: Construction of trial wave Up: Three-dimensional wave functions Previous: Trial wave function of   Contents
G.E. Astrakharchik 15-th of December 2004