phd-thesis/sec/app/tensor_wave.tex

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}