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Local kinetic energy and the drift force

In this section we will find the expression of the local kinetic energy

$\displaystyle T^{loc}({{\vec r_1},...,{\vec r}_N}) = -\frac{\hbar^2}{2m}\frac{\Delta\Psi({{\vec r_1},...,{\vec r}_N})}{\Psi({{\vec r_1},...,{\vec r}_N})}$     (2.124)

Let us calculate the second derivative in two steps, as the first derivative is important for the calculation of the drift force. We consider the Bijl-Jastrow form (2.37) of the trial wave function and will express the final results in terms of one- and two- body Bijl-Jastrow terms $f_1$ and $f_2$. The gradient of the many-body trial wave function is given by

$\displaystyle \vec\nabla_{{\vec r}_i}\Psi({{\vec r_1},...,{\vec r}_N})=\Psi({{\...
...\vert)}\frac{{{\vec r}_i-{\vec r}_k}}{\vert{{\vec r}_i-{\vec r}_k}\vert}\right)$     (2.125)

The full expression for the Laplacian is

$\displaystyle \Delta_{{\vec r}_i}\Psi({{\vec r_1},...,{\vec r}_N})\qquad=\qquad...
...t)}\frac{{{\vec r}_i-{\vec r}_k}}{\vert{{\vec r}_i-{\vec r}_k}\vert}
\right)^2+$      
$\displaystyle +\Psi({{\vec r_1},...,{\vec r}_N})\left(
\frac{f_1''({\vec r}_i)}...
...ec r}_k}\vert)}{f_2(\vert{{\vec r}_i-{\vec r}_k}\vert)}\right)^2
\right]\right)$     (2.126)

The kinetic energy can be written in a compact form

$\displaystyle T^{loc}({{\vec r_1},...,{\vec r}_N}) = \frac{\hbar^2}{2m}\left\{\...
...-\sum\limits_{i=1}^N \vert\vec F_i({{\vec r_1},...,{\vec r}_N})\vert^2\right\},$     (2.127)

where we introduced notation for the one- and two- body contribution to the local energy (see, also, (2.40))
$\displaystyle {\cal E}^{loc}_1(\vec r)$ $\textstyle =$ $\displaystyle -\frac{f_1''(\vec r)}{f_1(\vec r)}+\left(\frac{f_1'(\vec r)}{f_1(\vec r)}\right)^2$ (2.128)
$\displaystyle {\cal E}^{loc}_2(r)$ $\textstyle =$ $\displaystyle -\frac{f_2''(r)}{f_2(r)}+\left(\frac{f_2'(r)}{f_2(r)}\right)^2$ (2.129)

and introduced the drift force (see (2.39))
$\displaystyle \vec F_i({{\vec r_1},...,{\vec r}_N}) =
\frac{f_1'({\vec r}_i)}{f...
...c r}_k}\vert)}\frac{{{\vec r}_i-{\vec r}_k}}{\vert{{\vec r}_i-{\vec r}_k}\vert}$     (2.130)


next up previous contents
Next: Exponentiation Up: Local energy Previous: Local energy   Contents
G.E. Astrakharchik 15-th of December 2004