# CritPt / Challenge_47_main

task_id: 7e3c544c-7278-53eb-ab9c-d810c34d63a2
task_key: train--Challenge~5f47~5fmain
task_revision_id: 1

{"code_template":"def answer():\n    r\"\"\"\n    Return the value of $\\mathrm{Tr}(L^4)$ under the given spin configuration.\n\n    Inputs\n    ----------\n    None\n\n    Outputs\n    ----------\n    Tr_L4: float, value of $\\mathrm{Tr}(L^4)$ for the spin configuration\n        $$\\vec{m}(x) = ( \\sin \\theta \\cos \\phi,\\, \\sin \\theta \\sin \\phi,\\, \\cos \\theta )$$\n    \"\"\"\n\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\n    Tr_L4 = ...\n    # ---------------------------------------------------------------\n\n    return Tr_L4","problem_description":"# Problem setup:\nLet $\\vec m(x,t)$ be a classical spin field defined for $x\\in\\mathbb R$ and $t\\in[0,\\infty)$. The field satisfies the unit-length constraint\n\\begin{equation}\n\\vec m\\cdot\\vec m=1.\n\\end{equation}\nLet $\\sigma_\\alpha$ ($\\alpha=1,2,3$) denote the Pauli matrices, and define the $2\\times2$ matrix-valued field\n\\begin{equation}\nm=\\vec m\\cdot\\vec\\sigma= \\sum_{\\alpha = 1}^3 m^\\alpha\\sigma_\\alpha.\n\\end{equation}\nThe Lax operator is given by\n\\begin{equation}\nL=[\\mathcal H,m].\n\\end{equation}\nIts action should be understood analogously to that of a quantum operator.  Specifically, when acting on any $2\\times2$ matrix field $n(x)$, we have\n\\begin{equation}\nL(n)=\\mathcal H(mn)-m\\,\\mathcal H(n),\n\\end{equation}\nwhere $\\mathcal H$ is the Hilbert transform. For a scalar function $f(x)$, the Hilbert transform is defined as\n\\begin{equation}\n\\mathcal H[f(x)]=\\frac{\\mathrm P}{\\pi}\\int_{-\\infty}^{\\infty}\\frac{f(y)}{x-y}\\,dy=\\frac1{\\pi x}\\!*f(x).\n\\end{equation}\nWe may write either $\\mathcal H(f)$ or $f_{\\mathcal H}$ for this quantity. When $\\mathcal H$ acts on a matrix field, it is understood component-wise. We assume the boundary condition\n\\begin{equation}\n\\vec m(x\\to\\pm\\infty)=\\text{constant},\n\\end{equation}\ni.e., the spin field converges to a fixed vector as $x\\to\\pm\\infty$.\n\nThe trace $\\mathrm{Tr}(\\cdot)$ includes both the $2\\times2$ matrix indices and the spatial integral:\n\\begin{equation}\n\\mathrm{Tr}(\\cdot)=\\int_{-\\infty}^{\\infty}dx\\,\\mathrm{tr}(\\cdot).\n\\end{equation}\n\n# Main problem:\n\nCompute\n\\begin{equation}\n\\mathrm{Tr}(L^4),\n\\end{equation}\nfor the following spin configuration:\n\\begin{equation}\n\\vec{m}(x) = ( \\sin \\theta \\cos \\phi,\\, \\sin \\theta \\sin \\phi,\\, \\cos \\theta ),\n\\end{equation}\nwhich is a wave packet parameterized by\n\\begin{equation}\n\\theta(x) = x, \\qquad \\phi(x) = \\frac{2\\pi}{3}e^{-x^2}.\n\\end{equation}\n\nEvaluate this quantity to at least six decimal places."}

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GET /api/v1/write?intent=publish&task_id=7e3c544c-7278-53eb-ab9c-d810c34d63a2&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}