From a36c755ca1b7c925fbcde7dc24eeb910f773a77c Mon Sep 17 00:00:00 2001 From: hackbard Date: Sun, 5 Feb 2012 21:34:18 +0100 Subject: [PATCH] more detailed --- physics_compact/solid.tex | 16 ++++++++++++---- 1 file changed, 12 insertions(+), 4 deletions(-) diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index 3cb0480..30e078e 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -25,14 +25,22 @@ H=T+V+U\text{ ,} where \begin{eqnarray} T & = & \langle\Psi|\sum_{i=1}^N\frac{-\hbar^2}{2m}\nabla_i^2|\Psi\rangle\\ - & = & \sum_{i=1}^N \int d\vec{r} d\vec{r}' \, + & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \, \langle \Psi | \vec{r} \rangle \langle \vec{r} | - \frac{-\hbar^2}{2m}\nabla_i^2 + \nabla_i^2 | \vec{r}' \rangle \langle \vec{r}' | \Psi \rangle\\ + & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \, + \langle \Psi | \vec{r} \rangle \nabla_{\vec{r}_i} + \langle \vec{r} | \vec{r}' \rangle + \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\ + & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \, + \nabla_{\vec{r}_i} \langle \Psi | \vec{r} \rangle + \delta_{\vec{r}\vec{r}'} + \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\ & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} \, - \nabla_i \Psi^*(\vec{r}) \nabla_i \Psi(\vec{r}) + \nabla_{\vec{r}_i} \Psi^*(\vec{r}) \nabla_{\vec{r}_i} \Psi(\vec{r}) \text{ ,} \\ -V & = & V(\vec{r})\Psi^*(\vec{r})\Psi(\vec{r})d\vec{r} \text{ ,} \\ +V & = & \int V(\vec{r})\Psi^*(\vec{r})\Psi(\vec{r})d\vec{r} \text{ ,} \\ U & = & \frac{1}{2}\int\frac{1}{\left|\vec{r}-\vec{r}'\right|} \Psi^*(\vec{r})\Psi^*(\vec{r}')\Psi(\vec{r}')\Psi(\vec{r}) d\vec{r}d\vec{r}' -- 2.20.1