{"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":"0f0a7e02-ffb4-54cd-ae1d-0b7b13d7285b","task_key":"train--Challenge~5f29~5fmain","task_revision_id":"1","upstream_id":"Challenge_29_main","short_description":"Fermionic atoms trapped in an optical lattice are used to simulate the Hubbard…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\nimport sympy.physics.quantum.constants\\n\\nlambda_ = sp.symbols('lambda')\\nE       = sp.symbols('E')\\nW       = sp.symbols('W')\\nalpha   = sp.symbols('alpha')\\nm       = sp.symbols('m')\\na_s     = sp.symbols('a_s')\\nhbar    = sp.physics.quantum.constants.hbar\\n\\ndef answer(lambda_, E, W, alpha, m, a_s, hbar):\\n    r\\\"\\\"\\\"\\n    Return the expressions of tunneling energy t and contact interaction U in Sympy format.\\n\\n    Inputs\\n    ----------\\n    lambda_:      sympy.Symbol, wavelength, $\\\\lambda$\\n    E      :      sympy.Symbol, beam amplitude, $E$\\n    W      :      sympy.Symbol, beam waist, $W$\\n    alpha  :      sympy.Symbol, atomic polarizability, $\\\\alpha$\\n    m      :      sympy.Symbol, atomic mass, $m$\\n    a_s    :      sympy.Symbol, s-wave scattering length, $a_s$\\n    hbar   :      sympy.Symbol, reduced Planck constant, $\\\\hbar$\\n\\n    Outputs\\n    ----------\\n    t      :      sympy.Expr, the expression of tunneling energy t associated with the Hubbard model\\n    U      :      sympy.Expr, the expression of contact interaction U associated with the Hubbard model\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    t = ...  # a SymPy expression for the tunneling energy\\n    U = ...  # a SymPy expression for the contact interaction\\n    # ---------------------------------------------------------------\\n\\n    return t, U\",\"problem_description\":\"# Problem setup:\\n\\nFermionic atoms trapped in an optical lattice are used to simulate the Hubbard model. Specifically, two pairs of counter-propagating beams intersect at 90 degrees. The beams have equal wavelength $\\\\lambda$, amplitude $E$, beam waists $W$, and are polarized normal to the plane in which they intersect. The atoms are trapped in the resulting interference pattern because of their polarizability $\\\\alpha$ at a wavelength of $\\\\lambda$. The atoms have a mass $m$ and an $s$-wave scattering length of $a_s$.\\n\\n# Main problem:\\n\\nCompute the tunneling energy $t$ and contact interaction $U$ associated with the resulting Hubbard model.\\n\\nTo obtain analytical answers, assume that the Wannier functions can be approximated by the harmonic-oscillator eigenstates and that $W\\\\gg\\\\lambda$ and $V_0\\\\gg E_R$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}