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Hard-rod wave function (approximate)

The transformation (2.47) makes the implementation of the calculation quite sophisticated. Instead one can construct an approximate wave function in the same spirit as it will be done in the subsequent sections, i.e. by using the exact solution for the two particle scattering problem (1.60).

Using the solution (1.80) we propose

$\displaystyle f_2(z) =
\left\{
{\begin{array}{ll}
\displaystyle 0,& \vert z\ver...
...D}\vert))\right\vert,& \vert z\vert > \vert a_{1D}\vert\\
\end{array}}
\right.$     (2.48)

The advantage of this wave function is that it is solution of a two-body problem, so the interaction energy is always constant, which is much easier to sample numerically. The force and local energy can be easily obtained from $1D$ hard-sphere wave function. The difference is that ${\cal E}$ is a variational parameter here.

The drift force contribution (2.39) is given by

$\displaystyle {{\cal F}_2}(z) =
\left\{
{\begin{array}{ll}
\displaystyle 0,& \v...
...-\vert a_{1D}\vert), & \vert z\vert > \vert a_{1D}\vert\\
\end{array}}
\right.$     (2.49)

The $1D$ local energy (2.40) is

$\displaystyle {\cal E}^{loc}_2(z) =
\left\{
{\begin{array}{ll}
\displaystyle 0,...
...\vert a_{1D}\vert)), & \vert z\vert > \vert a_{1D}\vert\\
\end{array}}
\right.$     (2.50)


next up previous contents
Next: Wave function of the Up: One-dimensional wave functions Previous: Hard-rod wave function (exact)   Contents
G.E. Astrakharchik 15-th of December 2004