Derivatives of the densities

(a)
Divergence of tensor spin current density

$\displaystyle [\bbox{\nabla} \cdot \bbox{J}_m(\bbox{r})] =$ $\displaystyle \frac{1}{2i}\sum_{kss'tt'}\biggr\{ \bbox{\nabla} V_k(\bbox{r} s't')\cdot (\bbox{\nabla} \times \hat{\bbox{\sigma}}_{s's})V_k^\ast(\bbox{r} st)$    
  $\displaystyle -\bbox{\nabla} V_k^\ast(\bbox{r} st)\cdot(\bbox{\nabla} \times \hat{\bbox{\sigma}}_{s's})V_k(\bbox{r} s't')\biggr\}\tau_{t't}^m$    
  $\displaystyle \hspace*{-1cm}=\sum_{tt'} \biggr( DJ_r^{tt'}(rz) +
 DJ_\phi^{tt'}(rz) + DJ_z^{tt'}(rz) \biggr) \tau_{t't}^{m},$ (50)

where

$\displaystyle DJ_{r}^{tt'}(rz) =$ $\displaystyle \frac{1}{2}\sum_{k}\biggr\{-\frac{\Lambda^{-}}{r}[\partial_r V^{+}_{kt'}]V^{+\ast}_{kt}-\frac{\Lambda^{-}}{r}[\partial_r V^{+\ast}_{kt}]V^{+}_{kt'}$    
  $\displaystyle +[\partial_r V^{+\ast}_{kt}][\partial_z V^{-}_{kt'}]-[\partial_r V^{-\ast}_{kt}][\partial_z V^{+}_{kt'}]$    
  $\displaystyle -[\partial_r V^{-}_{kt'}][\partial_z V^{+\ast}_{kt}]+[\partial_r V^{+}_{kt'}][\partial_z V^{-\ast}_{kt}]$    
  $\displaystyle +\frac{\Lambda^{+}}{r}[\partial_r V^{-}_{kt'}]V^{-\ast}_{kt}+\frac{\Lambda^{+}}{r}[\partial_r V^{-\ast}_{kt}]V^{-}_{kt'}\biggr\},$ (51)
$\displaystyle DJ_{\phi}^{tt'}(rz) =$ $\displaystyle -\frac{1}{2}\sum_{k}\biggr\{\frac{\Lambda^{-}}{r}V^{+}_{kt'}[\partial_r V^{+\ast}_{kt}]+\frac{\Lambda^{-}}{r}V^{+\ast}_{kt}[\partial_r V^{+}_{kt'}]$    
  $\displaystyle -\frac{\Lambda^{-}}{r}V^{+}_{kt'}[\partial_z V^{-\ast}_{kt}]-\frac{\Lambda^{+}}{r}V^{-\ast}_{kt}[\partial_z V^{+}_{kt'}]$    
  $\displaystyle -\frac{\Lambda^{+}}{r}V^{-}_{kt'}[\partial_z V^{+\ast}_{kt}]-\frac{\Lambda^{-}}{r}V^{+\ast}_{kt}[\partial_z V^{-}_{kt'}]$    
  $\displaystyle -\frac{\Lambda^{+}}{r}V^{-}_{kt'}[\partial_r V^{-\ast}_{kt}]-\frac{\Lambda^{+}}{r}V^{-\ast}_{kt}[\partial_r V^{-}_{kt'}]\biggr\},$ (52)
$\displaystyle DJ_{z}^{tt'}(rz) =$ $\displaystyle - \frac{1} {2} \sum_{k}\biggr\{[\partial_z V^{+}_{kt'}][\partial_r V^{-\ast}_{kt}]-\frac{\Lambda^{+}}{r}[\partial_z V^{+}_{kt'}]V^{-\ast}_{kt}$    
  $\displaystyle -[\partial_z V^{-\ast}_{kt}][\partial_r V^{+}_{kt'}]-\frac{\Lambda^{-}}{r}[\partial_z V^{-\ast}_{kt}] V^{+}_{kt'}$    
  $\displaystyle -[\partial_z V^{-}_{kt'}][\partial_r V^{+\ast}_{kt}]-\frac{\Lambda^{-}}{r}[\partial_z V^{-}_{kt'}]V^{+\ast}_{kt}$    
  $\displaystyle +[\partial_z V^{+\ast}_{kt}][\partial_r V^{-}_{kt'}]-\frac{\Lambda^{+}}{r}[\partial_z V^{+\ast}_{kt}]V^{-}_{kt'}\biggr\}.$ (53)

Isospin components are given by

$\displaystyle \bbox{\nabla} \cdot \bbox{J}_0(\bbox{r}) =$ $\displaystyle \sum_{i=(r,\phi,z)} \biggr( DJ_{i}^{nn}(rz)+ DJ_{i}^{pp}(rz)
 \biggr), $ (54)
$\displaystyle \bbox{\nabla} \cdot \bbox{J}_1(\bbox{r}) =$ $\displaystyle \sum_{i=(r,\phi,z)}\biggr( DJ_{i}^{np}(rz)+
 DJ_{i}^{pn}(rz)\biggr), $ (55)
$\displaystyle \bbox{\nabla} \cdot \bbox{J}_2(\bbox{r}) =$ $\displaystyle i\sum_{i=(r,\phi,z)} \biggr(
 DJ_{i}^{np}(rz)-DJ_{i}^{pn}(rz)\biggr), $ (56)
$\displaystyle \bbox{\nabla} \cdot \bbox{J}_3(\bbox{r}) =$ $\displaystyle \sum_{i=(r,\phi,z)} \biggr( DJ_{i}^{nn}(rz)- DJ_{i}^{pp}(rz)\biggr).$ (57)

(b)
Curl of current density

$\displaystyle \bbox{\nabla} \times \bbox{j}_m(\bbox{r}) =\sum_{tt'} (\bbox{\nabla} \times \bbox{j})^{tt'}(rz) \tau_{t't}^{m},$ (58)

where

$\displaystyle (\bbox{\nabla} \times \bbox{j})^{tt'}(rz) =$ $\displaystyle \sum_{k}\biggr\{$    
  $\displaystyle \bbox{e}_r\biggr(\frac{\Lambda^{-}}{r}V^{+\ast}_{kt}[\partial_zV^{+}_{kt'}]+ \frac{\Lambda^{+}}{r}V^{-\ast}_{kt}[\partial_z V^{-}_{kt'}]$    
  $\displaystyle +\frac{\Lambda^{-}}{r}[\partial_z V^{+\ast}_{kt}]V^{+}_{kt'}+ \frac{\Lambda^{+}}{r}[\partial_z V^{-\ast}_{kt}]V^{-}_{kt'}\biggr)$    
  $\displaystyle +i\bbox{e}_{\phi}\biggr([\partial_z V^{+\ast}_{kt}][\partial_r V^{+}_{kt'}]+ [\partial_z V^{-\ast}_{kt}][\partial_r V^{-}_{kt'}]$    
  $\displaystyle -[\partial_r V^{+\ast}_{kt}][\partial_z V^{+}_{kt'}]-[\partial_r V^{-\ast}_{kt}][\partial_z V^{-}_{kt'}]\biggr)$    
  $\displaystyle -\bbox{e}_z\biggr(\frac{\Lambda^{-}}{r}[\partial_r V^{+\ast}_{kt}]V^{+}_{kt'}+ \frac{\Lambda^{+}}{r}[\partial_r V^{-\ast}_{kt}]V^{-}_{kt'}$    
  $\displaystyle +\frac{\Lambda^{-}}{r}V^{+\ast}_{kt}[\partial_r V^{+}_{kt'}]+ \frac{\Lambda^{+}}{r}V^{-\ast}_{kt}[\partial_r V^{-}_{kt'}]\biggr)\biggr\}$ (59)

(c)
Curl of spin density

$\displaystyle \bbox{\nabla} \times \bbox{s}_m(\bbox{r}) =\sum_{tt'} (\bbox{\nabla} \times \bbox{s})^{tt'}(rz)~\tau_{t't}^{m},$ (60)

where

$\displaystyle (\bbox{\nabla} \times \bbox{s})^{tt'}(rz) =$ $\displaystyle i \sum_{k}\biggr\{ \bbox{e}_r\biggr([\partial_z V^{-\ast}_{kt}]V^{+}_{kt'}+ V^{-\ast}_{kt}[\partial_z V^{+}_{kt'}]$    
  $\displaystyle -[\partial_z V^{+\ast}_{kt}]V^{-}_{kt'}- V^{+\ast}_{kt}[\partial_z V^{-}_{kt'}]\biggr)$    
  $\displaystyle -i \bbox{e}_{\phi}\biggr([\partial_z V^{-\ast}_{kt}]V^{+}_{kt'}+ V^{-\ast}_{kt}[\partial_z V^{+}_{kt'}]$    
  $\displaystyle +[\partial_z V^{+\ast}_{kt}]V^{-}_{kt'}+V^{+\ast}_{kt}[\partial_z V^{-}_{kt'}]$    
  $\displaystyle -[\partial_r V^{+\ast}_{kt}]V^{+}_{kt'}- V^{+\ast}_{kt}[\partial_r V^{+}_{kt'}]$    
  $\displaystyle +[\partial_r V^{-\ast}_{kt}]V^{-}_{kt'}+V^{-\ast}_{kt}[\partial_r V^{-}_{kt'}]\biggr)$    
  $\displaystyle +\bbox{e}_z\biggr([\partial_r V^{+\ast}_{kt}]V^{-}_{kt'}+ V^{+\ast}_{kt}[\partial_r V^{-}_{kt'}]$    
  $\displaystyle -[\partial_r V^{-\ast}_{kt}]V^{+}_{kt'}- V^{-\ast}_{kt}[\partial_r V^{+}_{kt'}]\biggr)\biggr\}.$ (61)

(d)
Laplacian of $ \rho$

$\displaystyle \nabla^{2}\rho_m(\bbox{r}) = \sum_{tt'} \biggr( 2 \tau^{tt'}(rz) + L^{tt'}(rz)\biggr)~~~\tau_{t't}^{m},$ (62)

where

$\displaystyle L^{tt'}(rz) =$ $\displaystyle \sum_{k}\biggr\{ \bar{V}^{+\ast}_{kt} [\nabla^{2} \bar{V}^{+}_{kt'} ] +
 \bar{V}^{-\ast}_{kt} [\nabla^{2} \bar{V}^{-}_{kt'}]$    
  $\displaystyle + \bar{V}^{+}_{kt'} [\nabla^{2}\bar{V}^{+\ast}_{kt} ]
 +\bar{V}^{-}_{kt'} [\nabla^{2} \bar{V}^{-\ast}_{kt} ]\biggr\},$ (63)
$\displaystyle \bar{V}^{+}_{kt} =$ $\displaystyle V^{+}_{kt} e^{i \Lambda^{-} \phi},$ (64)
$\displaystyle \bar{V}^{-}_{kt} =$ $\displaystyle V^{-}_{kt} e^{i \Lambda^{+} \phi}.$ (65)

(e)
Laplacian of $ \bbox{s}$

$\displaystyle \nabla^{2}\bbox{s}_m(\bbox{r}) =$ $\displaystyle \sum_{tt'} \biggr( 2 \bbox{T}^{tt'}(rz) + \bbox{S}^{tt'}(rz)$    
  $\displaystyle - \frac {{\bbox{e}}_r} { r^2 } s_r^{tt'}(rz) - \frac {{\bbox{e}}_\phi} {r^2} s_\phi^{tt'} (rz) \biggr)\tau_{t't}^{m},$ (66)

where

$\displaystyle \bbox{S}^{tt'}(rz) =$ $\displaystyle \sum_{k}\biggr\{ {\bbox{e}}_r \biggr( [\nabla^{2} \bar{V}^{-\ast}_{kt}] \bar{V}^{+}_{kt'} +\bar{V}^{-\ast}_{kt} [\nabla^{2} \bar{V}^{+}_{kt'}]$    
  $\displaystyle +[\nabla^{2} \bar{V}^{+\ast}_{kt} ] \bar{V}^{-}_{kt'} + \bar{V}^{+\ast}_{kt} [\nabla^{2} \bar{V}^{-}_{kt'} ]\biggr)$    
  $\displaystyle + i {\bbox{e}}_\phi \biggr( -[\nabla^{2} \bar{V}^{-\ast}_{kt}] \bar{V}^{+}_{kt'} - \bar{V}^{-\ast}_{kt} [\nabla^{2} \bar{V}^{+}_{kt'} ]$    
  $\displaystyle +[\nabla^{2} \bar{V}^{+\ast}_{kt} ] \bar{V}^{-}_{kt'} + \bar{V}^{+\ast}_{kt} [\nabla^{2} \bar{V}^{-}_{kt'} ]\biggr)$    
  $\displaystyle + {\bbox{e}}_z \biggr( [\nabla^{2} \bar{V}^{+\ast}_{kt} ] \bar{V}^{+}_{kt'} + \bar{V}^{+\ast}_{kt} [\nabla^{2} \bar{V}^{+}_{kt'} ]$    
  $\displaystyle -[\nabla^{2} \bar{V}^{-\ast}_{kt} ]\bar{V}^{-}_{kt'}- \bar{V}^{-\ast}_{kt} [\nabla^{2} \bar{V}^{-}_{kt'}]\biggr) \biggr\}.$ (67)

Jacek Dobaczewski 2014-12-07