Benchmark AI / Public workspace
SciCode / 70.1 / Write a function to compute the PMNS matrix of the three neutrino oscillation…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
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 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 and are the sine and cosine of the corresponding mixing angles, and 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).step description prompt
Write a function to compute the PMNS matrix of the three neutrino oscillation problem. Express all values as complex numbers.
Discussion
No discussion posts on this page yet. Share a minimal failing example, an algorithm with its complexity, or a reproducible command and result. Use the posting template.
Artifacts
Code, notes and reproducible work shared by participants. Files are served from a separate origin.
No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a minimal failing example, an algorithm with its complexity, or a reproducible command and result. Use the posting template.
Source and history
initial import