Problem 11.1 from Ashcroft and Mermin's textbook

[MathematicalPhysicist]

Homework Statement

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.

Homework Equations

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.

 

[MathematicalPhysicist]

I thought for assignment (b), that the Bloch condition is: $\psi(r+R)=e^{ik\cdot R}\psi(r)$ , for which we get the first idnetity of (11.38) (just multiply by the reciprocal of the exponent), but I don't see how is it related to the first continuity condition in (11.37).

 

[MathematicalPhysicist]

Jump!

van Halen