Morse potential

(0.001 seconds)

11—20 of 39 matching pages

11: 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

…

12: 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

…

13: 29.19 Physical Applications

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

14: 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). …

15: Bibliography Y

… ►

K. Yang and M. de Llano (1989) 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

…

16: 13.28 Physical Applications

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

17: 16.24 Physical Applications

… ►They are also potentially useful for the solution of more complicated restricted lattice walk problems, and the 3D Ising model; see Barber and Ninham (1970, pp. 147–148). …

18: 19.18 Derivatives and Differential Equations

… ►The next four differential equations apply to the complete case of

R_{F}

and

R_{G}

in the form

R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)

(see (19.16.20) and (19.16.23)). … ►Similarly, the function

u=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};x+iy,x-iy\right)

satisfies an equation of axially symmetric potential theory: …

19: 23.21 Physical Applications

… ►In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form

(1-x^{2})(1-k^{2}x^{2})

. The Weierstrass function

\wp

plays a similar role for cubic potentials in canonical form

g_{3}+g_{2}x-4x^{3}

. …

20: 9.16 Physical Applications

… ►Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). …