repulsive

SearchAdvancedHelp

(0.001 seconds)

1—10 of 11 matching pages

1: 33.22 Particle Scattering and Atomic and Molecular Spectra

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

Attractive potentials:	, .
…
Repulsive potentials:	$Z_1{Z}_2>0$, $\eta >0$.

… ► $R_{\mathrm{\infty }}=m_{\mathrm{e}}c{\alpha }^2/(2\mathrm{\hslash })$. … ►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 penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity $\rho /(F_{\mathrm{\ell }}^2(\eta ,\rho )+G_{\mathrm{\ell }}^2(\eta ,\rho ))$ (Mott and Massey (1956, pp. 63–65)). …

2: Bibliography Y

… ►

Coulomb wave functions in repulsive fields. Phys. Rev. 49 (2), pp. 174–189.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§33.10(i), §33.2(ii), §33.2(iii), §33.2(iv), §33.5(i), §33.5(iv), §33.9(ii).

Encodings:

BibTeX

…

3: 18.40 Methods of Computation

… ►Equation (18.40.7) provides step-histogram approximations to ${\int }_a^{x}d\mu (x)$, as shown in Figure 18.40.1 for $N=12$ and $120$, shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. ►

►►

Figure 18.40.1: Histogram approximations to the Repulsive Coulomb–Pollaczek, RCP, weight function integrated over $[-1,x)$, see Figure 18.39.2 for an exact result, for $Z=+1$, shown for $N=12$ and $N=120$. Magnify

… ►The example chosen is inversion from the ${\alpha }_n,{\beta }_n$ for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50). … ►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). …

4: 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. …

5: 18.39 Applications in the Physical Sciences

… ►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. … ►For $Z>0$ these are the repulsive CP OP’s with $x\in [-1,1]$ corresponding to the continuous spectrum of $\mathscr{H}(Z)$, $ϵ\in (0,\mathrm{\infty })$, and for we have the attractive CP OP’s, where the spectrum is complemented by the infinite set of bound state eigenvalues for fixed $l$. … ►Given that $a=-b$ in both the attractive and repulsive cases, the expression for the absolutely continuous, $x\in [-1,1]$, part of the function of (18.35.6) may be simplified: … ►The weight functions for both the attractive and repulsive cases are now unit normalized, see Bank and Ismail (1985), and Ismail (2009). …

6: Bibliography O

… ►

A closed form solution of the $s$-wave Bethe-Goldstone equation with an infinite repulsive core. J. Math. Phys. 27 (4), pp. 1154–1158.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§30.13(v).

Encodings:

BibTeX

…

7: Bibliography R

… ►

Computation of electron repulsion integrals using the Rys quadrature method. J. Comput. Chem. 4 (2), pp. 154–175.

ⓘ

External Links:

Link

Referenced by:

§18.32, §18.39(iii).

Encodings:

BibTeX

8: Bibliography H

… ►

Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Internat. J. Quantum Chem. 88 (6), pp. 701–734.

ⓘ

External Links:

Document

Referenced by:

§8.24(iii).

Encodings:

BibTeX

…

9: Bibliography B

… ►

Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.

ⓘ

External Links:

Document

Referenced by:

§29.19(i).

Encodings:

BibTeX

…

10: Bibliography C

… ►

Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity. Phys. Rev. A 62 (063610), pp. 1–10.

ⓘ

External Links:

Document

Referenced by:

§22.19(iii).

Encodings:

BibTeX

…