Derivatives of the densities

(a)

Divergence of tensor spin current density

$$[\bbox{\nabla} \cdot \bbox{J}_m(\bbox{r})] = \frac{1}{2i}\sum_{kss'tt'}\biggr\{ \bbox{\nabla} V_k(\bbox{r} s't')\cdot (\bbox{\nabla} \times \hat{\bbox{\sigma}}_{s's})V_k^\ast(\bbox{r} st)$$

$$-\bbox{\nabla} V_k^\ast(\bbox{r} st)\cdot(\bbox{\nabla} \times \hat{\bbox{\sigma}}_{s's})V_k(\bbox{r} s't')\biggr\}\tau_{t't}^m$$

$$\hspace*{-1cm}=\sum_{tt'} \biggr( DJ_r^{tt'}(rz) + DJ_\phi^{tt'}(rz) + DJ_z^{tt'}(rz) \biggr) \tau_{t't}^{m},$$ (50)

where

$$DJ_{r}^{tt'}(rz) = \frac{1}{2}\sum_{k}\biggr\{-\frac{\Lambda^{-}}{r}[\partial_r V^{+}_{kt'}]V^{+\ast}_{kt}-\frac{\Lambda^{-}}{r}[\partial_r V^{+\ast}_{kt}]V^{+}_{kt'}$$

$$+[\partial_r V^{+\ast}_{kt}][\partial_z V^{-}_{kt'}]-[\partial_r V^{-\ast}_{kt}][\partial_z V^{+}_{kt'}]$$

$$-[\partial_r V^{-}_{kt'}][\partial_z V^{+\ast}_{kt}]+[\partial_r V^{+}_{kt'}][\partial_z V^{-\ast}_{kt}]$$

$$+\frac{\Lambda^{+}}{r}[\partial_r V^{-}_{kt'}]V^{-\ast}_{kt}+\frac{\Lambda^{+}}{r}[\partial_r V^{-\ast}_{kt}]V^{-}_{kt'}\biggr\},$$ (51)

$$DJ_{\phi}^{tt'}(rz) = -\frac{1}{2}\sum_{k}\biggr\{\frac{\Lambda^{-}}{r}V^{+}_{kt'}[\partial_r V^{+\ast}_{kt}]+\frac{\Lambda^{-}}{r}V^{+\ast}_{kt}[\partial_r V^{+}_{kt'}]$$

$$-\frac{\Lambda^{-}}{r}V^{+}_{kt'}[\partial_z V^{-\ast}_{kt}]-\frac{\Lambda^{+}}{r}V^{-\ast}_{kt}[\partial_z V^{+}_{kt'}]$$

$$-\frac{\Lambda^{+}}{r}V^{-}_{kt'}[\partial_z V^{+\ast}_{kt}]-\frac{\Lambda^{-}}{r}V^{+\ast}_{kt}[\partial_z V^{-}_{kt'}]$$

$$-\frac{\Lambda^{+}}{r}V^{-}_{kt'}[\partial_r V^{-\ast}_{kt}]-\frac{\Lambda^{+}}{r}V^{-\ast}_{kt}[\partial_r V^{-}_{kt'}]\biggr\},$$ (52)

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

$$-[\partial_z V^{-\ast}_{kt}][\partial_r V^{+}_{kt'}]-\frac{\Lambda^{-}}{r}[\partial_z V^{-\ast}_{kt}] V^{+}_{kt'}$$

$$-[\partial_z V^{-}_{kt'}][\partial_r V^{+\ast}_{kt}]-\frac{\Lambda^{-}}{r}[\partial_z V^{-}_{kt'}]V^{+\ast}_{kt}$$

$$+[\partial_z V^{+\ast}_{kt}][\partial_r V^{-}_{kt'}]-\frac{\Lambda^{+}}{r}[\partial_z V^{+\ast}_{kt}]V^{-}_{kt'}\biggr\}.$$ (53)

Isospin components are given by

$$\bbox{\nabla} \cdot \bbox{J}_0(\bbox{r}) = \sum_{i=(r,\phi,z)} \biggr( DJ_{i}^{nn}(rz)+ DJ_{i}^{pp}(rz) \biggr),$$ (54)

$$\bbox{\nabla} \cdot \bbox{J}_1(\bbox{r}) = \sum_{i=(r,\phi,z)}\biggr( DJ_{i}^{np}(rz)+ DJ_{i}^{pn}(rz)\biggr),$$ (55)

$$\bbox{\nabla} \cdot \bbox{J}_2(\bbox{r}) = i\sum_{i=(r,\phi,z)} \biggr( DJ_{i}^{np}(rz)-DJ_{i}^{pn}(rz)\biggr),$$ (56)

$$\bbox{\nabla} \cdot \bbox{J}_3(\bbox{r}) = \sum_{i=(r,\phi,z)} \biggr( DJ_{i}^{nn}(rz)- DJ_{i}^{pp}(rz)\biggr).$$ (57)

(b)

Curl of current density

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

where

$$(\bbox{\nabla} \times \bbox{j})^{tt'}(rz) = \sum_{k}\biggr\{$$

$$\bbox{e}_r\biggr(\frac{\Lambda^{-}}{r}V^{+\ast}_{kt}[\partial_zV^{+}_{kt'}]+ \frac{\Lambda^{+}}{r}V^{-\ast}_{kt}[\partial_z V^{-}_{kt'}]$$

$$+\frac{\Lambda^{-}}{r}[\partial_z V^{+\ast}_{kt}]V^{+}_{kt'}+ \frac{\Lambda^{+}}{r}[\partial_z V^{-\ast}_{kt}]V^{-}_{kt'}\biggr)$$

$$+i\bbox{e}_{\phi}\biggr([\partial_z V^{+\ast}_{kt}][\partial_r V^{+}_{kt'}]+ [\partial_z V^{-\ast}_{kt}][\partial_r V^{-}_{kt'}]$$

$$-[\partial_r V^{+\ast}_{kt}][\partial_z V^{+}_{kt'}]-[\partial_r V^{-\ast}_{kt}][\partial_z V^{-}_{kt'}]\biggr)$$

$$-\bbox{e}_z\biggr(\frac{\Lambda^{-}}{r}[\partial_r V^{+\ast}_{kt}]V^{+}_{kt'}+ \frac{\Lambda^{+}}{r}[\partial_r V^{-\ast}_{kt}]V^{-}_{kt'}$$

$$+\frac{\Lambda^{-}}{r}V^{+\ast}_{kt}[\partial_r V^{+}_{kt'}]+ \frac{\Lambda^{+}}{r}V^{-\ast}_{kt}[\partial_r V^{-}_{kt'}]\biggr)\biggr\}$$ (59)

(c)

Curl of spin density

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

where

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

$$-[\partial_z V^{+\ast}_{kt}]V^{-}_{kt'}- V^{+\ast}_{kt}[\partial_z V^{-}_{kt'}]\biggr)$$

$$-i \bbox{e}_{\phi}\biggr([\partial_z V^{-\ast}_{kt}]V^{+}_{kt'}+ V^{-\ast}_{kt}[\partial_z V^{+}_{kt'}]$$

$$+[\partial_z V^{+\ast}_{kt}]V^{-}_{kt'}+V^{+\ast}_{kt}[\partial_z V^{-}_{kt'}]$$

$$-[\partial_r V^{+\ast}_{kt}]V^{+}_{kt'}- V^{+\ast}_{kt}[\partial_r V^{+}_{kt'}]$$

$$+[\partial_r V^{-\ast}_{kt}]V^{-}_{kt'}+V^{-\ast}_{kt}[\partial_r V^{-}_{kt'}]\biggr)$$

$$+\bbox{e}_z\biggr([\partial_r V^{+\ast}_{kt}]V^{-}_{kt'}+ V^{+\ast}_{kt}[\partial_r V^{-}_{kt'}]$$

$$-[\partial_r V^{-\ast}_{kt}]V^{+}_{kt'}- V^{-\ast}_{kt}[\partial_r V^{+}_{kt'}]\biggr)\biggr\}.$$ (61)

(d)

Laplacian of $\rho$

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

where

$$L^{tt'}(rz) = \sum_{k}\biggr\{ \bar{V}^{+\ast}_{kt} [\nabla^{2} \bar{V}^{+}_{kt'} ] + \bar{V}^{-\ast}_{kt} [\nabla^{2} \bar{V}^{-}_{kt'}]$$

$$+ \bar{V}^{+}_{kt'} [\nabla^{2}\bar{V}^{+\ast}_{kt} ] +\bar{V}^{-}_{kt'} [\nabla^{2} \bar{V}^{-\ast}_{kt} ]\biggr\},$$ (63)

$$\bar{V}^{+}_{kt} = V^{+}_{kt} e^{i \Lambda^{-} \phi},$$ (64)

$$\bar{V}^{-}_{kt} = V^{-}_{kt} e^{i \Lambda^{+} \phi}.$$ (65)

(e)

Laplacian of $\bbox{s}$

$$\nabla^{2}\bbox{s}_m(\bbox{r}) = \sum_{tt'} \biggr( 2 \bbox{T}^{tt'}(rz) + \bbox{S}^{tt'}(rz)$$

$$- \frac {{\bbox{e}}_r} { r^2 } s_r^{tt'}(rz) - \frac {{\bbox{e}}_\phi} {r^2} s_\phi^{tt'} (rz) \biggr)\tau_{t't}^{m},$$ (66)

where

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

$$+[\nabla^{2} \bar{V}^{+\ast}_{kt} ] \bar{V}^{-}_{kt'} + \bar{V}^{+\ast}_{kt} [\nabla^{2} \bar{V}^{-}_{kt'} ]\biggr)$$

$$+ i {\bbox{e}}_\phi \biggr( -[\nabla^{2} \bar{V}^{-\ast}_{kt}] \bar{V}^{+}_{kt'} - \bar{V}^{-\ast}_{kt} [\nabla^{2} \bar{V}^{+}_{kt'} ]$$

$$+[\nabla^{2} \bar{V}^{+\ast}_{kt} ] \bar{V}^{-}_{kt'} + \bar{V}^{+\ast}_{kt} [\nabla^{2} \bar{V}^{-}_{kt'} ]\biggr)$$

$$+ {\bbox{e}}_z \biggr( [\nabla^{2} \bar{V}^{+\ast}_{kt} ] \bar{V}^{+}_{kt'} + \bar{V}^{+\ast}_{kt} [\nabla^{2} \bar{V}^{+}_{kt'} ]$$

$$-[\nabla^{2} \bar{V}^{-\ast}_{kt} ]\bar{V}^{-}_{kt'}- \bar{V}^{-\ast}_{kt} [\nabla^{2} \bar{V}^{-}_{kt'}]\biggr) \biggr\}.$$ (67)