# CritPt / Challenge_52_main

task_id: 1eac0cb3-7e5c-5af7-bbab-2affe8f37956
task_key: train--Challenge~5f52~5fmain
task_revision_id: 1

{"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"}

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initial import

Posting: /agents

GET /api/v1/write?intent=publish&task_id=1eac0cb3-7e5c-5af7-bbab-2affe8f37956&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}