# SciCode / 70.7

task_id: 8c3efb97-155a-58b9-a06e-4bfb5fb70743
task_key: dev--49c68bb1-a088-53d9-a366-0885b5d1b63b--70~2e7
task_revision_id: 2

{"step_background":"Background\n\nThe 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 $\\mathbb{U}_{3}(L)=e^{-i h_{k} \\lambda^{k} L}$.\n\nWe wish to expand $\\mathbb{U}_{3}$ using an identity for the Gell-Mann matrices:\n\n$$\n\\begin{equation*}\n\\mathbb{U}_{3}(L)=u_{0} \\mathbb{1}+i u_{k} \\lambda^{k}\n\\end{equation*}\n$$\n\nwhere the complex coefficients $u_{0}$ and $u_{k}$ are functions of $L$ and the $h_{k}$.\n\nThe coefficients in the above equation can be written as $u_{0}=\\frac{1}{3} \\operatorname{Tr} \\mathbb{U}_{3}$ and $u_{k}=-\\frac{i}{2} \\operatorname{Tr}\\left(\\lambda^{k} \\mathbb{U}_{3}\\right)$\n\nAn application of Sylvester's formula to $3 \\times 3$ matrices allows us to express the coefficients in terms of the $\\mathrm{SU}(3)$ invariants\n\n$$\n\\begin{aligned}\n& L^{2}|h|^{2} \\equiv L^{2} h_{k} h^{k} \\\\\n& -L^{3}\\langle h\\rangle \\equiv-L^{3} d_{i j k} h^{i} h^{j} h^{k}\n\\end{aligned}\n$$\n\nNext, we solve the characteristic equation of $-h_{k} \\lambda^{k} L$, i.e., \n$$\\phi^{3}-\\left(L^{2}|h|^{2}\\right) \\phi-\\frac{2}{3}\\left(-L^{3}\\langle h\\rangle\\right)=0$$. \n\nThe equation follows from the Cayley-Hamilton theorem, written conveniently in terms of invariants $\\langle h\\rangle$ and $|h|^{2}$. Its three latent roots, or eigenvalues, are $\\phi_{m} \\equiv \\psi_{m} L(m=1,2,3)$, with\n\n$$\n\\begin{equation*}\n\\psi_{m} \\equiv \\frac{2|h|}{\\sqrt{3}} \\cos \\left[\\frac{1}{3}(\\chi+2 \\pi m)\\right]\n\\end{equation*}\n$$\n\nThe expansion coefficient $u_k$ can be expressed as:\n$$\n\\begin{align*}\n& u_{0}=\\frac{1}{3} \\sum_{m=1}^{3} e^{i L \\psi_{m}}\\\\\n& u_{k}=\\sum_{m=1}^{3} e^{i L \\psi_{m}} \\frac{\\psi_{m} h_{k}-(h * h)_{k}}{3 \\psi_{m}^{2}-|h|^{2}}\n\\end{align*}\n$$\n\nwhere $(h * h)_{i} \\equiv d_{i j k} h^{j} h^{k}$.","step_description_prompt":"Compute the expansion coefficients $u_k$ of the evolution operator of the three neutrino oscillation problem $\\mathbb{U}_{3}(L)=e^{-i h_{k} \\lambda^{k} L}$, if we expand it using the Gell-Mann matrices in the following way: $\\mathbb{U}_{3}(L)=u_{0} \\mathbb{1}+i u_{k} \\lambda^{k}$. Consider all coefficients from $k=0$ to $k=8$. $L$ is the baseline (i.e., the distance the neutrino has travelled in unit of eV)."}

Source: https://huggingface.co/datasets/SciCode1/SciCode

initial import

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GET /api/v1/write?intent=publish&task_id=8c3efb97-155a-58b9-a06e-4bfb5fb70743&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}