Coulomb potential barriers

SearchAdvancedHelp

(0.002 seconds)

1—10 of 78 matching pages

1: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►At positive energies $E>0$, $\rho \ge 0$, and: … ► $R_{\mathrm{\infty }}=m_{\mathrm{e}}c{\alpha }^2/(2\mathrm{\hslash })$. … ► … ►

§33.22(vi) Solutions Inside the Turning Point

►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: 17.17 Physical Applications

… ►See Kassel (1995). … ►It involves $q$-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …

3: 15.18 Physical Applications

… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). ►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).

4: Bibliography Y

… ►

$J$-matrix method: Extensions to arbitrary angular momentum and to Coulomb scattering. J. Math. Phys. 16, pp. 410–420.

ⓘ

Referenced by:

§18.39(iv).

Encodings:

BibTeX

►

$L$-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

ⓘ

Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).

Encodings:

BibTeX

… ►

Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.

ⓘ

External Links:

Document

Referenced by:

§1.18(viii).

Encodings:

BibTeX

… ►

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

…

5: 13.28 Physical Applications

… ►For potentials in quantum mechanics that are solvable in terms of confluent hypergeometric functions see Negro et al. (2000). … ►

§13.28(ii) Coulomb Functions

…

6: 33.17 Recurrence Relations and Derivatives

§33.17 Recurrence Relations and Derivatives

►

33.17.1 $$(\mathrm{\ell }+1)rf(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)f(ϵ,\mathrm{\ell };r)+\mathrm{\ell }\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r)=0,$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.2 $$(\mathrm{\ell }+1)\left(1+{\mathrm{\ell }}^2ϵ\right)rh(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)h(ϵ,\mathrm{\ell };r)+\mathrm{\ell }rh(ϵ,\mathrm{\ell }+1;r)=0,$$

ⓘ

Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.3 $$(\mathrm{\ell }+1)r{f}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)f(ϵ,\mathrm{\ell };r)-\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r),$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

►

33.17.4 $$(\mathrm{\ell }+1)r{h}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)h(ϵ,\mathrm{\ell };r)-rh(ϵ,\mathrm{\ell }+1;r).$$

ⓘ

Symbols:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

7: Bibliography S

… ►

Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

ⓘ

Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

§33.1, §33.14(iv), 9th item.

Encodings:

BibTeX

… ►

The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.

ⓘ

External Links:

Document

Referenced by:

§33.23(vii).

Encodings:

BibTeX

►

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

ⓘ

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

►

FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

ⓘ

Notes:

Single-precision Fortran code.

External Links:

Document, Code, ZentralBlatt

Referenced by:

10th item.

Encodings:

BibTeX

… ►

Perturbational results for diffraction of water-waves by nearly-vertical barriers. IMA J. Appl. Math. 34 (1), pp. 99–117.

ⓘ

External Links:

ISSN 0272-4960, Document, MathReview, ZentralBlatt

Referenced by:

§11.12.

Encodings:

BibTeX

…

8: 5.20 Physical Applications

… ►

Rutherford Scattering

►In nonrelativistic quantum mechanics, collisions between two charged particles are described with the aid of the Coulomb phase shift $\mathrm{ph}\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right)$; see (33.2.10) and Clark (1979). … ►Suppose the potential energy of a gas of $n$ point charges with positions $x_1,x_2,\mathrm{\dots },x_n$ and free to move on the infinite line , is given by ►

5.20.1

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $j$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $W$: potential energy

Permalink:

http://dlmf.nist.gov/5.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

… ►

5.20.2 $$P(x_1,\mathrm{\dots },x_n)=C\exp \left(-W/(kT)\right),$$

ⓘ

Symbols:

$\exp z$: exponential function, $n$: nonnegative integer, $x$: real variable, $W$: potential energy, $T$: temperature and $C$: constant

Permalink:

http://dlmf.nist.gov/5.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

…

9: 33.15 Graphics

§33.15 Graphics

►

§33.15(i) Line Graphs of the Coulomb Functions $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$

►

►►

Figure 33.15.1: $f(ϵ,\mathrm{\ell };r),h(ϵ,\mathrm{\ell };r)$ with $\mathrm{\ell }=0,ϵ=4$. Magnify

… ►

§33.15(ii) Surfaces of the Coulomb Functions $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, and $c(ϵ,\mathrm{\ell };r)$

… ►

►

►

Figure 33.15.9: $h(ϵ,\mathrm{\ell };r)$ with . Magnify 3D Help

…

10: 33.2 Definitions and Basic Properties

… ►

§33.2(i) Coulomb Wave Equation

… ►The functions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are defined by …${\sigma }_{\mathrm{\ell }}\left(\eta \right)$ is the Coulomb phase shift. … ►

§33.2(iv) Wronskians and Cross-Product

…