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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):
![$\displaystyle f_1(\r) = \exp\left\{-\frac{x^2+y^2}{2\alpha_\rho^2}-\frac{z^2}{2\alpha_z^2}\right\}$](img936.gif) |
|
|
(2.89) |
Here,
and
determine the Gaussian width of
in the
longitudinal and transverse direction, respectively. These variational parameters
and
are optimized in the course of the VMC calculation by
minimizing the energy expectation value. The two-body correlation factor
(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}](img939.gif) |
(2.90) |
The constraint
for
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}](img941.gif) |
(2.91) |
where
and
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
and
are determined by fitting the elliptical surface to
the nodal surface obtained by solving the Schrödinger equation for
,
Eqs. 4.7 and 4.8, by performing a B-spline basis set
calculation. In contrast to
, the constraint
in Eq. 2.91
parameterizes the many-body nodal surface for
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.
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G.E. Astrakharchik
15-th of December 2004