Physics Help Forum problem 11.1 from Ashcroft and Mermin.
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 Nov 12th 2018, 12:58 PM #1 Junior Member     Join Date: Aug 2009 Posts: 9 problem 11.1 from Ashcroft and Mermin. I asked my question in physicsforums, perhaps someone here knows how to solve it or can provide guidance. https://www.physicsforums.com/thread...xtbook.958929/ Thanks!
Nov 12th 2018, 02:16 PM   #2
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 Originally Posted by Alan I asked my question in physicsforums, perhaps someone here knows how to solve it or can provide guidance. https://www.physicsforums.com/thread...xtbook.958929/ Thanks!
Please post the original problem when posting between fora.

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 Nov 12th 2018, 04:57 PM #3 Junior Member   Join Date: Apr 2018 Posts: 20 I never knew that the plural form of "forum" is "fora" instead of "forums"... topsquark likes this.
 Nov 12th 2018, 11:01 PM #4 Junior Member     Join Date: Aug 2009 Posts: 9 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. topsquark likes this.
 Nov 14th 2018, 10:27 AM #5 Senior Member   Join Date: Apr 2015 Location: Somerset, England Posts: 1,035 There is only one question at the end of chapter 11 in Ashcroft_Mermin. Does the title of the chapter (Other methods) give you a clue? Which method would you choose? (did you understand Green's functions?) Attached Thumbnails
Nov 14th 2018, 10:39 AM   #6
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 Originally Posted by studiot There is only one question at the end of chapter 11 in Ashcroft_Mermin. Does the title of the chapter (Other methods) give you a clue? Which method would you choose? (did you understand Green's functions?)
There are 3 questions, and I don't understand how to start answering question 1.

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