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End of NBO

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-10-05 17:49:06 +02:00
parent 5a7583760e
commit 21cfa5051a
7 changed files with 2298 additions and 809 deletions
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292
sec/app/tensor_wave.tex Normal file
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For the sake of completeness we report the expression of the full \nbo tensor wave function.
In what follows $L = \frac{l}{k_+}$.
We have
\begin{equation}
\begin{split}
\mqty(
S_{u\, u}
\\
S_{u\, v}
\\
S_{u\, z}
\\
S_{u\, i}
\\
S_{v\, v}
\\
S_{v\, z}
\\
S_{v\, i}
\\
S_{z\, z}
\\
S_{z\, i}
\\
S_{i\, i}
)
& =
\Biggl\lbrace
\cS_{u\, u}
\mqty(
1
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
)\,
+
\cS_{u\, v}
\mqty(
\frac{i}{k_+\, u} + \frac{L^2}{\Delta^2\, u^2}
\\
1
\\
L
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
)\,
+
\cS_{u\, z}
\mqty(
\frac{2\, L}{\Delta\, u}
\\
0
\\
\Delta\, u
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
)\,
+
\cS_{u\, i}
\mqty(
0
\\
0
\\
0
\\
1
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
)\,
\\
& +
\cS_{v\, v}
\mqty(
-\frac{3}{4\, k_+^2\, u^2}
+
\frac{3\, i\, L^2}{2\, \Delta^2\, k_+\, u^3}
+
\frac{L^4}{4\, \Delta^4\, u^4}
\\
\frac{i}{2\, k_+\, u}
+
\frac{L^2}{2\, \Delta^2\, u^2}
\\
\frac{3\, i\, L}{2\, k_+\, u}
+
\frac{L^3}{2\, \Delta^2\, u^2}
\\
0
\\
1
\\
L
\\
0
\\
\frac{i\, \Delta^2\, u}{k_+}
+
L^2
\\
0
\\
0
\\
)\,
+
\cS_{v\, z}
\mqty(
\frac{3\, i\, L}{\Delta\, k_+\, u^2}
+
\frac{L^3}{\Delta^3\, u^3}
\\
\frac{L}{\Delta\, u}
\\
\frac{3\, L^2}{2\, \Delta\, u}
+
\frac{3\, i\, \Delta}{2\, k_+}
\\
0
\\
0
\\
\Delta\, u
\\
0
\\
2\, \Delta\, L\, u
\\
0
\\
0
\\
)\,
\\
& +
\cS_{v\, i}
\mqty(
0
\\
0
\\
0
\\
\frac{i}{2\, k_+\, u}
+
\frac{L^2}{2\, \Delta^2\, u^2}
\\
0
\\
0
\\
1
\\
0
\\
L
\\
0
\\
)\,
+
\cS_{z\, z}
\mqty(
\frac{i}{k_+\, u}
+
\frac{L^2}{\Delta^2\, u^2}
\\
0
\\
L
\\
0
\\
0
\\
0
\\
0
\\
\Delta^2\, u^2
\\
0
\\
0
\\
)\,
+
\cS_{z\, i}
\mqty(
0
\\
0
\\
0
\\
\frac{L}{\Delta\, u}
\\
0
\\
0
\\
0
\\
0
\\
\Delta\, u
\\
0
\\
)\,
\\
& +
\cS_{i\, j}
\mqty(
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
0
\\
\delta_{i j}
\\
)\,
\Biggr\rbrace
\phi_{\kmkr}.
\end{split}
\end{equation}