{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"1eac0cb3-7e5c-5af7-bbab-2affe8f37956","task_key":"train--Challenge~5f52~5fmain","task_revision_id":"1","upstream_id":"Challenge_52_main","short_description":"The Efimov effect is one of the few examples of a three-body problem that can be…","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the value of $P(s_1)$ to three decimal places.\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    P_s1: float, the overlap of the hyperangular parts of the non-interacting and Efimov wave functions for $s=s_1$, \\\\$P(s_1)\\\\$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    P_s1 = ...\\n    # ---------------------------------------------------------------\\n\\n    return P_s1\",\"problem_description\":\"# Problem setup:\\n\\nThe Efimov effect is one of the few examples of a three-body problem that can be solved exactly.  Let's calculate the the angular portion of the three-body wave function and its normalization constant for the Efimov effect.  \\n\\n Consider three identical bosons with spatial coordinates $\\\\vec{r}_1$, $\\\\vec{r}_2$, and $\\\\vec{r}_3$.  Define Jacobi vectors $\\\\vec{r}_{ij} = \\\\vec{r}_i - \\\\vec{r}_j$ and $\\\\vec{\\\\rho}_{k,ij} = (2\\\\vec{r}_k - \\\\vec{r}_i - \\\\vec{r}_j)/\\\\sqrt{3}$, where $(i,j,k)$ is a permutation of $(1,2,3)$.  The hyperradius is defined as $R^2 = (r_{ij}^2 + \\\\rho_{k,ij}^2)/2$.  We use notation $r_{ij}= |\\\\vec{r}_{ij}|$ and $\\\\rho_{ij}= |\\\\vec{\\\\rho}_{ij}|$. The hyperangle $\\\\alpha$ is defined as $r_{ij}=2R\\\\sin\\\\alpha_k$ and $\\\\rho_{k,ij}=\\\\sqrt{2}R\\\\cos\\\\alpha_k$ such that $\\\\alpha_k = \\\\arctan(r_{ij}/\\\\rho_{k,ij})$.  The range of the hyperangle is restricted between $0$ and $\\\\pi/2$ such that $r_{ij}$ and $\\\\rho_{k,ij}$ remain always positive.  We define the shorthand $\\\\alpha\\\\equiv \\\\alpha_3$ as well as permutation operators $\\\\hat{P}_{13}$ and $\\\\hat{P}_{23}$, where $\\\\hat{P}_{ij}$ is the permutation operator that swaps particle indices $i$ and $j$.\\n\\nConsider the hyperangular part of the three-body wave function for an Efimov state $\\\\phi(s,\\\\alpha) = (1 + \\\\hat{Q})F(s,\\\\alpha)/\\\\sqrt{N(s)}$, where $F(s,\\\\alpha) = \\\\varphi(s,\\\\alpha)/ \\\\sin(2\\\\alpha)$, $\\\\varphi(s,\\\\alpha) = \\\\sin(s(\\\\pi/ 2 - \\\\alpha))$, $\\\\hat{Q} = \\\\hat{P}_{13} + \\\\hat{P}_{23}$, and $N(s)$ is the normalization factor.\\n\\nThe values $s$ are obtained from solving\\n$$\\\\frac{d\\\\varphi(s,0)}{d\\\\alpha} + \\\\frac{8}{\\\\sqrt{3}}\\\\varphi(s,\\\\pi/3)=0.$$\\nAssume $s$ is a real number.  We are concerned with the first non-integer value of $s$ that solves this equation.  Calculate your answer to three significant decimal places and record the value in the variable $s_1$.\\n\\n\\nNext, calculate the overlap of the hyperangular parts of the non-interacting and Efimov wave functions for $s=s_1$.  This overlap is $P(s) = G(s)^2/(N(s)H)$ for $s=s_1$.  \\n\\nHere $N(s)$, $G(s)$, and $F(s)$ are overlap integrals given by:\\n\\n$N(s) = \\\\int_0^{\\\\pi/2} d\\\\alpha \\\\sin(2\\\\alpha)^2\\\\phi(s,\\\\alpha)^2$,\\n\\n$G(s) = \\\\int_0^{\\\\pi/2} d\\\\alpha \\\\sin(2\\\\alpha)^2\\\\phi(s,\\\\alpha)$,\\n\\n$H = \\\\int_0^{\\\\pi/2} d\\\\alpha \\\\sin(2\\\\alpha)^2$.\\n\\n\\n# Main problem:\\n\\nCalculate $N(s)$, $H$, and $G(s)$ and use these results to obtain $P(s_1)$ to three decimal places.  \\n\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}