more detailed

[lectures/latex.git] / physics_compact / solid.tex

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 3cb0480..30e078e 100644 (file)
--- 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}'