potential

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11: 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

…

12: 13.28 Physical Applications

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

13: 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). …

14: 18.39 Applications in the Physical Sciences

… ►defines the potential for a symmetric restoring force

-kx

for displacements from

x=0

. … ►argument b) The Morse Oscillator … ►c) A Rational SUSY Potential argument ►The Schrödinger equation with potential … ►Now use spherical coordinates (1.5.16) with

r

instead of

\rho

, and assume the potential

V

to be radial. …

15: 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: …

16: 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}

. …

17: 9.16 Physical Applications

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

18: 22.19 Physical Applications

… ►

22.19.4

\frac{{\mathrm{d}}^{2}x(t)}{{\mathrm{d}t}^{2}}=-\frac{\mathrm{d}V(x)}{\mathrm{% d}x},

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $V(x)$ : potential energy, $x(t)$ : coordinate and $t$ : time
Permalink:: http://dlmf.nist.gov/22.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.19(ii), §22.19 and Ch.22

►where

V(x)

is the potential energy, and

x(t)

is the coordinate as a function of time

t

. The potential ►

22.19.5

V(x)=\pm\tfrac{1}{2}x^{2}\pm\tfrac{1}{4}\beta x^{4}

ⓘ

Symbols:: $V(x)$ : potential energy, $x(t)$ : coordinate and $\beta$ : real positive
Permalink:: http://dlmf.nist.gov/22.19.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.19(ii), §22.19 and Ch.22

…

19: 10.73 Physical Applications

… ►Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … ►In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …

20: DLMF Project News

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