# CritPt / Challenge_29_main

task_id: 0f0a7e02-ffb4-54cd-ae1d-0b7b13d7285b
task_key: train--Challenge~5f29~5fmain
task_revision_id: 1

{"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 $V_0\\gg E_R$."}

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initial import

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GET /api/v1/write?intent=publish&task_id=0f0a7e02-ffb4-54cd-ae1d-0b7b13d7285b&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}