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Local Densities and Currents

The complete density matrix $\rho(\vec{r}\sigma t, \vec{r}'\sigma' t')$ in spin-isospin space as defined in (1) can be decomposed into the sum of scalar $\rho_{t t_3} (\vec{r}, \vec{r}')$ and vector densities $\vec{s}_{t t_3} (\vec{r}, \vec{r}')$, where the subscripts denote the isospin quantum numbers:
$\displaystyle {\rho(\vec{r}\sigma \tau, \vec{r}'\sigma' \tau')
}$
  = $\displaystyle {\textstyle\frac{{1}}{{4}}} \Big[
\rho_{00} (\vec{r}, \vec{r}') \...
...}') \cdot \mbox{{\boldmath {$\sigma$}}}_{\sigma \sigma'}
\, \delta_{\tau \tau'}$  
    $\displaystyle + \delta_{\sigma \sigma'} \! \sum_{t_3 = -1}^{+1}
\rho_{1 t_3} (\...
...dot \mbox{{\boldmath {$\sigma$}}}_{\sigma \sigma'}
\tau^{t_3}_{\tau\tau'}
\Big]$  

The quantities $\mbox{{\boldmath {$\sigma$}}}_{\sigma \sigma'}$ and $\tau^{t_3}_{\tau\tau'}$ are matrix elements of the Pauli matrices in spin and isospin space. In terms of these, the local density $\rho$, spin density $\vec{s}$, kinetic density $\tau$, kinetic spin density $\vec{T}$, current $\vec{j}$, and spin-orbit tensor $\stackrel{\leftrightarrow}{J}$ are
$\displaystyle \rho_{t t_3} (\vec{r})$ = $\displaystyle \rho_{t t_3} (\vec{r}, \vec{r}) \phantom{\Big\vert _{\vec{r}=\vec{r}'}}$  
$\displaystyle \vec{s}_{t t_3} (\vec{r})$ = $\displaystyle \vec{s}_{t t_3} (\vec{r}, \vec{r}) \phantom{\Big\vert _{\vec{r}=\vec{r}'}}$  
$\displaystyle \tau_{t t_3} (\vec{r})$ = $\displaystyle \nabla \cdot \nabla' \rho_{t t_3} (\vec{r},\vec{r}')
\Big\vert _{\vec{r}=\vec{r}'}$  
$\displaystyle \vec{T}_{t t_3} (\vec{r})$ = $\displaystyle \nabla \cdot \nabla'
\vec{s}_{t t_3} (\vec{r},\vec{r}') \Big\vert _{\vec{r}=\vec{r}'}$  
$\displaystyle \vec{j}_{t t_3} (\vec{r})$ = $\displaystyle - {\textstyle\frac{{\mbox{\rm\scriptsize {i}}}}{{2}}} ( \nabla - \nabla' )
\rho_{t t_3} (\vec{r},\vec{r}') \Big\vert _{\vec{r}=\vec{r}'}$  
$\displaystyle J_{t t_3, ij} (\vec{r})$ = $\displaystyle - {\textstyle\frac{{\mbox{\rm\scriptsize {i}}}}{{2}}} ( \nabla - ...
...la' )_i \;
s_{t t_3,j} (\vec{r},\vec{r}') \Big\vert _{\vec{r}=\vec{r}'}
\quad .$ (31)

The densities $\rho$, $\tau$, and $\stackrel{\leftrightarrow}{J}$ are time-even, while $\vec{s}$, $\vec{T}$, and $\vec{j}$ are time-odd. See [20] for a more detailed discussion.
next up previous
Next: Energy density functional from Up: Gamow-Teller strength and the Previous: Summary, conclusions, and outlook
Jacek Dobaczewski
2002-03-15