Momentum matrix elements: $\hat{O} = \nabla$

For $k_a \neq k_b$ we have for each component $j = 1 \dots d$ of the grad operator,

$$\left\langle a\right\vert\partial_j\left\vert b\right\rangl... ...k_a^2 a^* b - (\partial_m a^*) \partial_m b \right] \right\} .$$ (H.20)

This follows from row 1 of the matrix ${\mathcal{T}}^{-1}$. Summation over $m$ is implied.

For $k_a = k_b = k$ it is unambiguous to write all components together as a vector relation thus,

$$\left\langle a\right\vert\nabla\left\vert b\right\rangle _{... ...+ {\mathbf r} (a^* \partial_n b - b \partial_n a^* ) \right ].$$ (H.21)

This follows from the singular solution ${\mathcal{T}}^T {\mathbf \xi} = {\mathbf e}$ for the first unit vector ${\mathbf e}$. It is interesting that the equal wavenumber case involves one order of derivative less than the unequal case. In the Dirichlet BC case, (H.21) vanishes.