potential

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1: 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)).

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: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►At positive energies

E>0

\rho\geq 0

, and: … ►

R_{\infty}=m_{e}c{\alpha}^{2}/(2\hbar)

. … ►Both variable sets may be used for attractive and repulsive potentials: the

(\epsilon,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. … ►

§33.22(vi) Solutions Inside the Turning Point

…

4: 14.31 Other Applications

… ►

§14.31(ii) Conical Functions

… ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). …

5: Bibliography Q

… ►

C. Quesne (2011) Higher-Order SUSY, Exactly Solvable Potentials, and Exceptional Orthogonal Polynomials. Modern Physics Letters A 26, pp. 1843–1852.

ⓘ

External Links:: Document, Link
Referenced by:: §18.38(iii).
Encodings:: BibTeX

6: 5.20 Physical Applications

… ►Suppose the potential energy of a gas of

n

point charges with positions

x_{1},x_{2},\dots,x_{n}

and free to move on the infinite line

-\infty<x<\infty

, is given by ►

5.20.1

W=\frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell<j\leq n}\ln|x_{\ell% }-x_{j}|.

ⓘ

Symbols:: $\ln\NVar{z}$ : 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},\dots,x_{n})=C\exp\left(-W/(kT)\right),

ⓘ

Symbols:: $\exp\NVar{z}$ : 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

… ►

5.20.3

\psi_{n}(\beta)=\int_{{\mathbb{R}}^{n}}e^{-\beta W}\,\mathrm{d}x\\ =(2\pi)^{n/2}\beta^{-(n/2)-(\beta n(n-1)/4)}\*\left(\Gamma\left(1+\tfrac{1}{2}% \beta\right)\right)^{-n}\prod_{j=1}^{n}\Gamma\left(1+\tfrac{1}{2}j\beta\right).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\mathbb{R}$ : real line, $j$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $W$ : potential energy, $\psi_{n}(\beta)$ : partition function and $\beta=1/(kT)$
Permalink:: http://dlmf.nist.gov/5.20.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §5.20, §5.20 and Ch.5

… ►

5.20.4

W=-\sum_{1\leq\ell<j\leq n}\ln|e^{i\theta_{\ell}}-e^{i\theta_{j}}|,

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $j$ : nonnegative integer, $n$ : nonnegative integer and $W$ : potential energy
Permalink:: http://dlmf.nist.gov/5.20.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §5.20, §5.20 and Ch.5

…

7: 19.33 Triaxial Ellipsoids

… ►

§19.33(ii) Potential of a Charged Conducting Ellipsoid

… ►The potential is ►

19.33.5

V(\lambda)=QR_{F}\left(a^{2}+\lambda,b^{2}+\lambda,c^{2}+\lambda\right),

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $Q$ : charge and $V(\lambda)$ : potential
Permalink:: http://dlmf.nist.gov/19.33.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(ii), §19.33 and Ch.19

…

8: 29.19 Physical Applications

… ►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …

9: 12.17 Physical Applications

… ►For this topic and other boundary-value problems see Boyd (1973), Hillion (1997), Magnus (1941), Morse and Feshbach (1953a, b), Müller (1988), Ott (1985), Rice (1954), and Shanmugam (1978). ►Lastly, parabolic cylinder functions arise in the description of ultra cold atoms in harmonic trapping potentials; see Busch et al. (1998) and Edwards et al. (1999).

10: 15.19 Methods of Computation

… ►For fast computation of

F\left(a,b;c;z\right)

with

a, b

and

c

complex, and with application to Pöschl–Teller–Ginocchio potential wave functions, see Michel and Stoitsov (2008). …