Atomic and Solid State Physics Atomic and Solid State Physics Help Forum 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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 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!
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See the forum rules here.   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?)   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.  Tags 111, ashcroft, mermin, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Physics Forum Discussions Thread Thread Starter Forum Replies Last Post Alan Atomic and Solid State Physics 0 May 14th 2017 07:49 AM 