1. The problem statement, all variables and given/known data
Let ##\vec{r}## locate a point just within the boundary of a primitive cell ##C_0## and ##\vec{r}'## another point infinitesimally displaced from ##\vec{r}## just outside the same boundary. The continuity equations for ##\psi(\vec{r})## are:
$$ (11.37) \lim_{r\to r'} [\psi(\vec{r})\psi(\vec{r}')]=0$$
$$\lim_{r\to r'} [\nabla \psi(\vec{r})\nabla \psi(\vec{r}')]=0$$
(a) Verify that any point ##\vec{r}## on the surface of a primitive cell is separated by some Bravais lattice vector ##\vec{R}## from another surface point and that the normals to the cell at ##\vec{r}## and ##\vec{r}+\vec{R}## are oppositely directed.
(b) Using the fact that ##\psi## can be chosen to have the Bloch form, show that the continuity conditions can equally well be written in terms of the values of ##\psi## entirely withing a primitive cell:
$$(11.38) \psi(\vec{r}) = e^{i\vec{k}\cdot\vec{r}}\psi(\vec{r}+\vec{R})$$
$$\nabla \psi(\vec{r})= e^{i\vec{k}\cdot \vec{R}}\nabla \psi(\vec{r}+\vec{R})$$
for pairs of points on the surface separated by direct lattice vectors ##\vec{R}##.
(c) Show that the only information in the second of equations (11.38) not already contained in the first is in the equation:
$$(11.39)\hat{n}(\vec{r})\cdot \nabla \psi(\vec{r})=e^{i\vec{k}\cdot \vec{R}}\hat{n}(\vec{r}+\vec{R})\cdot \nabla \psi(\vec{r}+\vec{R}),$$
where the vector ##\hat{n}## is normal to the surface of the cell.
2. Relevant equations
3. The attempt at a solution
I am quite overwhelmed by this question, and am not sure where to start.
I would appreciate some guidance as to how to solve this problem.
Thanks.
