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1: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►At positive energies $E>0$, $\rho \ge 0$, and: ► ►►

Attractive potentials:	, .
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… ► $R_{\mathrm{\infty }}=m_{\mathrm{e}}c{\alpha }^2/(2\mathrm{\hslash })$. … ►Both variable sets may be used for attractive and repulsive potentials: the $(ϵ,r)$ set cannot be used for a zero potential because this would imply $r=0$ for all $s$, and the $(\eta ,\rho )$ set cannot be used for zero energy $E$ because this would imply $\rho =0$ always. … ►The functions ${\varphi }_{n,\mathrm{\ell }}(r)$ defined by (33.14.14) are the hydrogenic bound states in attractive Coulomb potentials; their polynomial components are often called associated Laguerre functions; see Christy and Duck (1961) and Bethe and Salpeter (1977). …

2: 33.1 Special Notation

… ►The main functions treated in this chapter are first the Coulomb radial functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, $c(ϵ,\mathrm{\ell };r)$ (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. …

3: 18.39 Applications in the Physical Sciences

… ►Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with $Z$ protons, involve Laguerre polynomials of fractional index. … ►As in this subsection both positive (repulsive) and negative (attractive) Coulomb interactions are discussed, the prefactor of $Z/r$ in (18.39.43) has been set to $+1$, rather than the $-1$ of (18.39.28) implying that is an attractive interaction, $Z>0$ being repulsive. … ►These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3). … ►In the attractive case (18.35.6_4) for the discrete parts of the weight function where with , are also simplified: …The weight functions for both the attractive and repulsive cases are now unit normalized, see Bank and Ismail (1985), and Ismail (2009). …

4: 18.40 Methods of Computation

… ►Further, exponential convergence in $N$, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty },\mathrm{\infty })$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

5: 3.6 Linear Difference Equations

… ►Difference equations are simple and attractive for computation. …

6: Bibliography B

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The attractive Coulomb potential polynomials. Constr. Approx. 1 (2), pp. 103–119.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (W. A. Al-Salam)

Referenced by:

§18.39(iv), §18.39(iv).

Encodings:

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7: Bibliography S

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Coulomb functions for attractive and repulsive potentials and for positive and negative energies. Comput. Phys. Comm. 146 (2), pp. 225–249.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§1.17(ii), §33.1, §33.14(i), §33.14(ii), §33.14(iii), §33.14(iv), §33.14(v), §33.16(i), §33.16(ii), §33.16(iii), §33.16(v), §33.17, §33.19, §33.20(i), §33.20(ii), §33.20(iii), §33.21(i), Ch.33.

Encodings:

BibTeX

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