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In this section we will find the expression of the local kinetic energy
![$\displaystyle T^{loc}({{\vec r_1},...,{\vec r}_N}) = -\frac{\hbar^2}{2m}\frac{\Delta\Psi({{\vec r_1},...,{\vec r}_N})}{\Psi({{\vec r_1},...,{\vec r}_N})}$](img1000.gif) |
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(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
and
. The gradient of the
many-body trial wave function is given by
![$\displaystyle \vec\nabla_{{\vec r}_i}\Psi({{\vec r_1},...,{\vec r}_N})=\Psi({{\...
...\vert)}\frac{{{\vec r}_i-{\vec r}_k}}{\vert{{\vec r}_i-{\vec r}_k}\vert}\right)$](img1003.gif) |
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(2.125) |
The full expression for the Laplacian is
![$\displaystyle \Delta_{{\vec r}_i}\Psi({{\vec r_1},...,{\vec r}_N})\qquad=\qquad...
...t)}\frac{{{\vec r}_i-{\vec r}_k}}{\vert{{\vec r}_i-{\vec r}_k}\vert}
\right)^2+$](img1004.gif) |
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![$\displaystyle +\Psi({{\vec r_1},...,{\vec r}_N})\left(
\frac{f_1''({\vec r}_i)}...
...ec r}_k}\vert)}{f_2(\vert{{\vec r}_i-{\vec r}_k}\vert)}\right)^2
\right]\right)$](img1005.gif) |
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(2.126) |
The kinetic energy can be written in a compact form
![$\displaystyle T^{loc}({{\vec r_1},...,{\vec r}_N}) = \frac{\hbar^2}{2m}\left\{\...
...-\sum\limits_{i=1}^N \vert\vec F_i({{\vec r_1},...,{\vec r}_N})\vert^2\right\},$](img1006.gif) |
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(2.127) |
where we introduced notation for the one- and two- body contribution to the local
energy (see, also, (2.40))
and introduced the drift force (see (2.39))
![$\displaystyle \vec F_i({{\vec r_1},...,{\vec r}_N}) =
\frac{f_1'({\vec r}_i)}{f...
...c r}_k}\vert)}\frac{{{\vec r}_i-{\vec r}_k}}{\vert{{\vec r}_i-{\vec r}_k}\vert}$](img1011.gif) |
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(2.130) |
Next: Exponentiation
Up: Local energy
Previous: Local energy
  Contents
G.E. Astrakharchik
15-th of December 2004