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SciCode / 70.8 / Write a function to compute the 3×3 evolution operator as defined in…

Problem

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step description prompt

Write a function to compute the 3×33 \times 3 evolution operator as defined in the previous prompt (). Then compute the oscillation probabilities of the three neutrino oscillation problem using the expression Pνα→νβ(L)=|νβ†U3(L)να|2P_{\nu_{\alpha} \rightarrow \nu_{\beta}}(L)=\left|\nu_{\beta}^{\dagger} U_{3}(L) \nu_{\alpha}\right|^{2}, where ν\nu is one of electron neutrino (ee), muon neutrino (μ\mu) or tau neutrino (τ\tau). Use $\nu_{e}=\left(\begin{array}{lll}1 & 0 & 0\end{array}\right)^{\mathrm{T}}, \nu_{\mu}=\left(\begin{array}{lll}0 & 1 & 0\end{array}\right)^{\mathrm{T}}$, and $\nu_{\tau}=\left(\begin{array}{lll}0 & 0 & 1\end{array}\right)^{\mathrm{T}}$. Return the probabilities in a list in the following order : e→e/μ/τe \to e/\mu/\tau, μ→e/μ/τ\mu \to e/\mu/\tau, and τ→e/μ/τ\tau \to e/\mu/\tau.
Plain-text mathematical notation (without MathML)
Write a function to compute the 3×3 evolution operator as defined in the previous prompt (). Then compute the oscillation probabilities of the three neutrino oscillation problem using the expression P_(ν_(α)→ν_(β))(L)=|ν_(β)^(†)U₃(L)ν_(α)|², where ν is one of electron neutrino (e), muon neutrino (μ) or tau neutrino (τ). Use $\nu_{e}=\left(\begin{array}{lll}1 & 0 & 0\end{array}\right)^{\mathrm{T}}, \nu_{\mu}=\left(\begin{array}{lll}0 & 1 & 0\end{array}\right)^{\mathrm{T}}$, and $\nu_{\tau}=\left(\begin{array}{lll}0 & 0 & 1\end{array}\right)^{\mathrm{T}}$. Return the probabilities in a list in the following order : e→e/μ/τ, μ→e/μ/τ, and τ→e/μ/τ.
Original LaTeX notation
Write a function to compute the $3 \times 3$ evolution operator as defined in the previous prompt (). Then compute the oscillation probabilities of the three neutrino oscillation problem using the expression $P_{\nu_{\alpha} \rightarrow \nu_{\beta}}(L)=\left|\nu_{\beta}^{\dagger} U_{3}(L) \nu_{\alpha}\right|^{2}$, where $\nu$ is one of electron neutrino ($e$), muon neutrino ($\mu$) or tau neutrino ($\tau$). Use $\nu_{e}=\left(\begin{array}{lll}1 & 0 & 0\end{array}\right)^{\mathrm{T}}, \nu_{\mu}=\left(\begin{array}{lll}0 & 1 & 0\end{array}\right)^{\mathrm{T}}$, and $\nu_{\tau}=\left(\begin{array}{lll}0 & 0 & 1\end{array}\right)^{\mathrm{T}}$. Return the probabilities in a list in the following order : $e \to e/\mu/\tau$, $\mu \to e/\mu/\tau$, and $\tau \to e/\mu/\tau$.

step background

Background The evolution operator is U3(L)=e−iH3L=\mathbb{U}_{3}(L)=e^{-i H_{3} L}= e−ih01Le−ihkλkLe^{-i h_{0} \mathbb{1} L} e^{-i h_{k} \lambda^{k} L}. If we discard the global phase, we are left with U3(L)=e−ihkλkL\mathbb{U}_{3}(L)=e^{-i h_{k} \lambda^{k} L}.
Plain-text mathematical notation (without MathML)
Background

The evolution operator is U₃(L)=e^(−iH₃L)= e^(−ih₀1L)e^(−ih_(k)λ^(k)L). If we discard the global phase, we are left with U₃(L)=e^(−ih_(k)λ^(k)L).
Original LaTeX notation
Background

The evolution operator is $\mathbb{U}_{3}(L)=e^{-i H_{3} L}=$ $e^{-i h_{0} \mathbb{1} L} e^{-i h_{k} \lambda^{k} L}$. If we discard the global phase, we are left with $\mathbb{U}_{3}(L)=e^{-i h_{k} \lambda^{k} L}$.

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Source and history

upstream_id / Original problem ID
70.8
task_id
b6d21d4a-3ffa-5df7-8647-f593157f1cc8
task_key
dev--49c68bb1-a088-53d9-a366-0885b5d1b63b--70~2e8
task_revision_id
5
config / split
/ dev

Official source

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