selection.json

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4104 bytes, as of the version from 2026-10-27 01:12 (current). Recorded here, not run.

{
  "tid": "T511",
  "kind": "coulombg",
  "rationale": "The free case eta=0 is an intrinsic degeneration. For the p, d and f partial waves ell=1,2,3, the centrifugal turning radius is sqrt(ell(ell+1)). The s-wave free case has no positive turning radius and is not included in that group. The handbook worked examples include ell=0,1,2,3 at eta=2, rho=5 as reproducibility benchmarks, not a measured or mathematically distinguished coupling. At that same reference coupling the outer turning radius is 2+sqrt(4+ell(ell+1)), intrinsic conditional on eta=2. A turning point means a zero of the coefficient in the reduced wave equation, not a stationary point of F or G. No nonzero eta is claimed physically fixed.",
  "unresolved": "The nonzero coupling eta=2 is a documented handbook benchmark only, not established external physical demand.",
  "sources": [
    "https://dlmf.nist.gov/33.2.E2",
    "https://dlmf.nist.gov/33.5",
    "https://dlmf.nist.gov/33.22",
    "https://personal.math.ubc.ca/~cbm/aands/page_543.htm"
  ],
  "rows": [
    {
      "params": {
        "ell": "0",
        "eta": "2",
        "rho": "4"
      },
      "argument": "turning",
      "reason": "Outer turning point for the partial wave and coupling in the handbook example. The radius is intrinsic conditional on that reference coupling."
    },
    {
      "params": {
        "ell": "0",
        "eta": "2",
        "rho": "5"
      },
      "argument": "5",
      "reason": "Abramowitz and Stegun page 543 worked examples 1 and 2; explicitly handbook benchmarks, not physical necessity."
    },
    {
      "params": {
        "ell": "1",
        "eta": "0",
        "rho": "sqrt(2)"
      },
      "argument": "turning",
      "label": "$\\sqrt{2}$",
      "reason": "Centrifugal turning point of the free partial wave; eta=0 is an intrinsic degeneration."
    },
    {
      "params": {
        "ell": "1",
        "eta": "2",
        "rho": "2+sqrt(6)"
      },
      "argument": "turning",
      "reason": "Outer turning point for the partial wave and coupling in the handbook example. The radius is intrinsic conditional on that reference coupling.",
      "label": "$2+\\sqrt{6}$"
    },
    {
      "params": {
        "ell": "1",
        "eta": "2",
        "rho": "5"
      },
      "argument": "5",
      "reason": "Abramowitz and Stegun page 543 worked examples 1 and 2; explicitly handbook benchmarks, not physical necessity."
    },
    {
      "params": {
        "ell": "2",
        "eta": "0",
        "rho": "sqrt(6)"
      },
      "argument": "turning",
      "label": "$\\sqrt{6}$",
      "reason": "Centrifugal turning point of the free partial wave; eta=0 is an intrinsic degeneration."
    },
    {
      "params": {
        "ell": "2",
        "eta": "2",
        "rho": "2+sqrt(10)"
      },
      "argument": "turning",
      "reason": "Outer turning point for the partial wave and coupling in the handbook example. The radius is intrinsic conditional on that reference coupling.",
      "label": "$2+\\sqrt{10}$"
    },
    {
      "params": {
        "ell": "2",
        "eta": "2",
        "rho": "5"
      },
      "argument": "5",
      "reason": "Abramowitz and Stegun page 543 worked examples 1 and 2; explicitly handbook benchmarks, not physical necessity."
    },
    {
      "params": {
        "ell": "3",
        "eta": "0",
        "rho": "sqrt(12)"
      },
      "argument": "turning",
      "label": "$\\sqrt{12}$",
      "reason": "Centrifugal turning point of the free partial wave; eta=0 is an intrinsic degeneration."
    },
    {
      "params": {
        "ell": "3",
        "eta": "2",
        "rho": "6"
      },
      "argument": "turning",
      "reason": "Outer turning point for the partial wave and coupling in the handbook example. The radius is intrinsic conditional on that reference coupling."
    },
    {
      "params": {
        "ell": "3",
        "eta": "2",
        "rho": "5"
      },
      "argument": "5",
      "reason": "Abramowitz and Stegun page 543 worked examples 1 and 2; explicitly handbook benchmarks, not physical necessity."
    }
  ]
}