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\n", " The momentum space Schrödinger equation is solved for bound states by converting the integral equation into a set of linear algebraic equations:\n", " $$\n", "\\frac{k^{2}}{2\\mu} \\psi(k) + \\frac{2}{\\pi} \\int_{0}^{\\infty} dp\n", "p^{2} V(k,p) \\psi(p) = E \\psi(k), \\quad\n", "V(k,p) = \\frac{1}{kp} \\int_{0}^{\\infty} dr\\sin(kr) V(r) \\sin(pr),\n", "\\\\\n", "\\int_{0}^{\\infty} dp p^{2} V(k,p) \\psi(p) \\simeq \\sum_{j=1}^{N}\n", "w_jk_{j}^{2} V(k,k_j) \\psi(k_j) \\quad \\Rightarrow \\quad\n", "\\frac{k^{2}}{2\\mu} \\psi(k) + \\frac{2}{\\pi} \\sum_{j=1}^{N}w_j\n", "k_{j}^{2} V(k,k_j) \\psi(k_j) =E. \\label{BS.pse3}\n", "$$ " ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/html": [ "
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