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author
hackbard
<hackbard@hackdaworld.org>
Sun, 5 Feb 2012 20:34:18 +0000
(21:34 +0100)
committer
hackbard
<hackbard@hackdaworld.org>
Sun, 5 Feb 2012 20:34:18 +0000
(21:34 +0100)
physics_compact/solid.tex
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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\\
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} |
\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\\
| \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} \,
& = & \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{ ,} \\
\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}'
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}'