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  "body": "\n## Summary\n\nCoulomb wave functions are solutions of the Coulomb wave equation $u'' + \\left( 1 - \\frac{2\\eta}{\\rho} - \\frac{L(L + 1)}{\\rho^2} \\right) u = 0$.\nThis equation is the radial part of the [[Schrödinger equation|EPZR6H2WJN]] for a charged particle in the field of a point charge.\nThe regular solution $F_L(\\eta, \\rho)$ vanishes at $\\rho = 0$, and the irregular solution $G_L(\\eta, \\rho)$ does not.\nBoth are [[confluent hypergeometric functions|6AZ4ZBN1M8]] in disguise.\n\n## Explanations\n\n### L1 human\n\nWithout the charge, that is for $\\eta = 0$, the equation describes a free wave, and for $L = 0$ its solutions are $\\sin \\rho$ and $\\cos \\rho$.\nThe term $2\\eta / \\rho$ adds the electric attraction or repulsion, and $L(L + 1)/\\rho^2$ adds the effect of angular momentum.\nFar from the origin the solutions still look like sine waves, but with a phase that drifts slowly like $\\eta \\log 2\\rho$.\nThe regular function $F_L$ generalises $\\sin \\rho$, and the irregular function $G_L$ generalises $\\cos \\rho$.\n",
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