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SciCode / 70.1 / Write a function to compute the PMNS matrix of the three neutrino oscillation…

Problem

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step description prompt

Write a function to compute the PMNS matrix of the three neutrino oscillation problem. Express all values as complex numbers.

step background

Background The flavour mixing matrix for the three neutrino families is $$ \begin{pmatrix} \nu_{e} \\ \nu_{\mu} \\ \nu_{\tau} \end{pmatrix} = \begin{pmatrix} U_{e1} & U_{e2} & U_{e3} \\ U_{\mu1} & U_{\mu2} & U_{\mu3} \\ U_{\tau1} & U_{\tau2} & U_{\tau3} \end{pmatrix} \cdot \begin{pmatrix} \nu_{1} \\ \nu_{2} \\ \nu_{3} \end{pmatrix} $$ The PMNS matrix UU can be expressed as: $$ U = \begin{pmatrix} c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta_{CP}} \\ -s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i\delta_{CP}} & c_{12}c_{23} - s_{12}s_{23}s_{13}e^{i\delta_{CP}} & s_{23}c_{13} \\ s_{12}s_{23} - c_{12}c_{23}s_{13}e^{i\delta_{CP}} & -c_{12}s_{23} - s_{12}c_{23}s_{13}e^{i\delta_{CP}} & c_{23}c_{13} \end{pmatrix} $$ where sijs_{ij} and cijc_{ij} are the sine and cosine of the corresponding mixing angles, and δcp\delta_{cp} is the CP-violating phase (in radians).
Plain-text mathematical notation (without MathML)
Background
The flavour mixing matrix for the three neutrino families is


$$
\begin{pmatrix}
   \nu_{e} \\
   \nu_{\mu}  \\
   \nu_{\tau}
\end{pmatrix}
=
\begin{pmatrix}
   U_{e1} & U_{e2} & U_{e3} \\
   U_{\mu1} & U_{\mu2} & U_{\mu3} \\
   U_{\tau1} & U_{\tau2} & U_{\tau3}
\end{pmatrix}
\cdot
\begin{pmatrix}
   \nu_{1} \\
   \nu_{2}  \\
   \nu_{3}
\end{pmatrix}
$$

The PMNS matrix U can be expressed as:
$$
U = \begin{pmatrix}
c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta_{CP}} \\
-s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i\delta_{CP}} & c_{12}c_{23} - s_{12}s_{23}s_{13}e^{i\delta_{CP}} & s_{23}c_{13} \\
s_{12}s_{23} - c_{12}c_{23}s_{13}e^{i\delta_{CP}} & -c_{12}s_{23} - s_{12}c_{23}s_{13}e^{i\delta_{CP}} & c_{23}c_{13}
\end{pmatrix}
$$

where s_(ij) and c_(ij) are the sine and cosine of the corresponding mixing angles, and δ_(cp) is the CP-violating phase (in radians).
Original LaTeX notation
Background
The flavour mixing matrix for the three neutrino families is


$$
\begin{pmatrix}
   \nu_{e} \\
   \nu_{\mu}  \\
   \nu_{\tau}
\end{pmatrix}
=
\begin{pmatrix}
   U_{e1} & U_{e2} & U_{e3} \\
   U_{\mu1} & U_{\mu2} & U_{\mu3} \\
   U_{\tau1} & U_{\tau2} & U_{\tau3}
\end{pmatrix}
\cdot
\begin{pmatrix}
   \nu_{1} \\
   \nu_{2}  \\
   \nu_{3}
\end{pmatrix}
$$

The PMNS matrix $U$ can be expressed as:
$$
U = \begin{pmatrix}
c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta_{CP}} \\
-s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i\delta_{CP}} & c_{12}c_{23} - s_{12}s_{23}s_{13}e^{i\delta_{CP}} & s_{23}c_{13} \\
s_{12}s_{23} - c_{12}c_{23}s_{13}e^{i\delta_{CP}} & -c_{12}s_{23} - s_{12}c_{23}s_{13}e^{i\delta_{CP}} & c_{23}c_{13}
\end{pmatrix}
$$

where $s_{ij}$ and $c_{ij}$ are the sine and cosine of the corresponding mixing angles, and $\delta_{cp}$ is the CP-violating phase (in radians).

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Source and history

upstream_id / Original problem ID
70.1
task_id
adbd390d-1702-57e7-bd92-c77e8ca00a02
task_key
dev--49c68bb1-a088-53d9-a366-0885b5d1b63b--70~2e1
task_revision_id
5
config / split
/ dev

Official source

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