Inverse kinematics - Revision history

193.175.86.132 at 15:09, 22 October 2014

2014-10-22T15:09:26Z

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	~~In [[Particle physics\|particle]] and [[atomic physics]], a '''Yukawa potential''' (also called a '''screened [[Coulomb potential]]''') is a [[potential]] of the form~~

	:~~<math>V_\text{Yukawa}(r)= -g^2\frac{e^{-kmr}}{r},<~~/~~math>~~

	~~where ''g'' is a magnitude scaling constant, i~~.e. is the amplitude of potential, ''m'' is the mass of the affected particle, ''r'' is the radial distance to the particle, and ''k'' is another scaling constant, which finally the product of ''km'' is the inverse scope. ~~The potential is~~ [~~[Monotonic function\|monotone increasing]], [[Force#Potential_energy\|implying]] that the force is always attractive~~.

	~~The [[Coulomb potential]] of [[electromagnetism]] is an example of a Yukawa potential with e<sup>−''kmr''<~~/~~sup> equal to 1 everywhere; this is taken to mean that the [[photon]] mass ''m'' is equal to 0~~.

	~~In interactions between a [[meson]] field and a [[fermion]] field, the constant ''g'' is equal to the [[coupling constant]] between those fields~~. ~~In the case of the [[nuclear force]], the fermions would be a [[proton]] and another proton or a [[neutron]]~~.

	~~== History =~~=
	~~[[Hideki Yukawa~~]~~] showed in the 1930s that such a potential arises from the exchange of a massive [[scalar field (quantum field theory)\|scalar field]] such as the field of the [[pion]] whose mass is~~ <~~math~~>m<~~/math~~>~~. Since the field mediator is massive the corresponding force has a certain range, which is inversely proportional to the mass.~~<~~ref name="martinshaw"~~>~~{{cite book \|url=~~http://~~books~~.~~google~~.fr/~~books?id=whIbrWJdEJQC&pg=PA18 \|title=Particle Physics \|author1=Brian Robert Martin \|author2=Graham Shaw \|page=18 \|year=2008 }}<~~/~~ref>~~

	~~==Relation to Coulomb potential==~~
	~~[[File:Yukawa m compare~~.~~svg\|thumb\|'''Figure 1:''' A comparison of Yukawa potentials where g=1 and with various values for m.]~~]
	~~[[File:Yukawa coulomb compare.svg\|thumb\|'''Figure~~ 2~~:''' A "long~~-~~range" comparison of Yukawa and Coulomb potentials' strengths where g=1.]]~~

	If the ~~mass is zero (i~~.e.~~, m=0), then the Yukawa potential equals~~ a ~~Coulomb potential~~, ~~and~~ the ~~range is said to~~ be ~~infinite~~.
	~~In fact, we have:~~

	~~:<math>m=0 \Rightarrow e^{-m r}= e^0 = 1~~.~~</math>~~
	~~Consequently~~, the ~~equation~~
	~~:<math>V_{\text{Yukawa}}(r)= -g^2 \;\frac{e^{-mr}}{r}</math>~~
	~~simplifies to~~ the ~~form of the Coulomb potential~~
	~~:<math>V_{\text{Coulomb}}(r)= -g^2 \;\frac{1}{r}~~.~~</math>~~

	~~A comparison~~ of ~~the long range potential strength~~ for ~~Yukawa~~ and ~~Coulomb is shown in Figure 2~~. ~~It can be seen that~~ the ~~Coulomb potential has effect over a~~ greater ~~distance whereas~~ the ~~Yukawa potential approaches zero rather quickly~~. ~~However~~, ~~any Yukawa potential or Coulomb potential~~ are ~~non-zero for any large r.~~

	~~==Fourier transform==~~
	~~The easiest way to understand that~~ the ~~Yukawa potential is associated with a massive field is by examining its [[Fourier transform]]~~. ~~One has~~

	:<~~math~~>~~V(\mathbf{r})=\frac{-g^2}{(2\pi)^3} \int e^{i\mathbf{k \cdot r}}~~
	~~\frac {4\pi}{k^2+m^2} \;d^3k~~<~~/math~~>

	~~where the integral is performed over all possible values of the 3-vector momentum ''k''. In this form, the fraction~~ <~~math~~>~~4\pi/(k^2+m^2)~~<~~/math~~> ~~is seen~~ to be the ~~[[propagator]] or [[Green's function]]~~ of ~~the~~ [~~[Klein-Gordon equation]]~~.

	==~~Feynman amplitude=~~=
	~~[[Image:1pxchg.svg\|right\|Single particle exchange~~]]
	~~The Yukawa potential can~~ be ~~derived as the lowest order amplitude of the interaction of~~ a ~~pair of fermions. The [[Yukawa interaction]] couples~~ the ~~fermion field <math>\psi(x~~)~~</math>~~ to ~~the meson field <math>\phi(x)</math> with the coupling term~~

	~~:<math>\mathcal{L}_\mathrm{int}(x) = g\overline{\psi}(x)\phi(x) \psi(x)~~.~~</math>~~

	~~The [[scattering amplitude]] for two fermions, one with initial momentum <math>p_1</math> and~~ the ~~other with momentum <math>p_2</math>~~, ~~exchanging~~ a ~~meson~~ with ~~momentum ''k''~~, is ~~given by the [[Feynman diagram]] on the right.~~

	~~The Feynman rules for each vertex associate a factor~~ of ~~''g'' with~~ the ~~amplitude; since this diagram~~ has ~~two vertices, the total amplitude will have~~ a ~~factor~~ of ~~<math>g^2</math>~~. ~~The line in the middle~~, ~~connecting~~ the ~~two fermion lines, represents the exchange of a meson. The Feynman rule for a particle exchange is to use the propagator; the propagator for~~ a ~~massive meson is <math>-4\pi/(k^2+m^2)</math>. Thus, we see~~ that ~~the Feynman amplitude for this graph is nothing more than~~

	~~:<math>V(\mathbf{k})=-g^2\frac{4\pi}{k^2+m^2}.</math>~~

	~~From the previous section~~, ~~this is seen to~~ be ~~the Fourier transform~~ of ~~the Yukawa potential.~~
	~~{{clr}}~~

	~~==See also==~~
	*[[Yukawa interaction]]
	*[[Screened Poisson equation]]

	~~==References==~~
	~~===Citations===~~
	~~{{reflist}}~~

	~~===Texts===~~
	* [[Hideki Yukawa\|H. Yukawa]], ~~''On~~ the ~~interaction of elementary particles''. (1935) Proc. Phys. Math. Soc. Japan. '''17''' 48~~

	* [[Gerald Edward Brown]] and A. ~~D. Jackson, ''The Nucleon-Nucleon Interaction'', (1976) North-Holland Publishing, Amsterdam ISBN 0-7204-0335-9~~

	~~[[Category:Particle physics]]~~
	~~[[Category:Quantum field theory]]~~
	~~[[Category:Potentials]]~~

en>Dratman: /* Inverse kinematics and 3D animation */ verb tense

2013-12-27T13:42:08Z

Inverse kinematics and 3D animation: verb tense

New page

In [[Particle physics|particle]] and [[atomic physics]], a '''Yukawa potential''' (also called a '''screened [[Coulomb potential]]''') is a [[potential]] of the form

:<math>V_\text{Yukawa}(r)= -g^2\frac{e^{-kmr}}{r},</math>

where ''g'' is a magnitude scaling constant, i.e. is the amplitude of potential, ''m'' is the mass of the affected particle, ''r'' is the radial distance to the particle, and ''k'' is another scaling constant, which finally the product of ''km'' is the inverse scope. The potential is [[Monotonic function|monotone increasing]], [[Force#Potential_energy|implying]] that the force is always attractive.

The [[Coulomb potential]] of [[electromagnetism]] is an example of a Yukawa potential with e<sup>−''kmr''</sup> equal to 1 everywhere; this is taken to mean that the [[photon]] mass ''m'' is equal to 0.

In interactions between a [[meson]] field and a [[fermion]] field, the constant ''g'' is equal to the [[coupling constant]] between those fields. In the case of the [[nuclear force]], the fermions would be a [[proton]] and another proton or a [[neutron]].

== History ==
[[Hideki Yukawa]] showed in the 1930s that such a potential arises from the exchange of a massive [[scalar field (quantum field theory)|scalar field]] such as the field of the [[pion]] whose mass is <math>m</math>. Since the field mediator is massive the corresponding force has a certain range, which is inversely proportional to the mass.<ref name="martinshaw">{{cite book |url=http://books.google.fr/books?id=whIbrWJdEJQC&pg=PA18 |title=Particle Physics |author1=Brian Robert Martin |author2=Graham Shaw |page=18 |year=2008 }}</ref>

==Relation to Coulomb potential==
[[File:Yukawa m compare.svg|thumb|'''Figure 1:''' A comparison of Yukawa potentials where g=1 and with various values for m.]]
[[File:Yukawa coulomb compare.svg|thumb|'''Figure 2:''' A "long-range" comparison of Yukawa and Coulomb potentials' strengths where g=1.]]

If the mass is zero (i.e., m=0), then the Yukawa potential equals a Coulomb potential, and the range is said to be infinite.
In fact, we have:

:<math>m=0 \Rightarrow e^{-m r}= e^0 = 1.</math>
Consequently, the equation
:<math>V_{\text{Yukawa}}(r)= -g^2 \;\frac{e^{-mr}}{r}</math>
simplifies to the form of the Coulomb potential
:<math>V_{\text{Coulomb}}(r)= -g^2 \;\frac{1}{r}.</math>

A comparison of the long range potential strength for Yukawa and Coulomb is shown in Figure 2. It can be seen that the Coulomb potential has effect over a greater distance whereas the Yukawa potential approaches zero rather quickly. However, any Yukawa potential or Coulomb potential are non-zero for any large r.

==Fourier transform==
The easiest way to understand that the Yukawa potential is associated with a massive field is by examining its [[Fourier transform]]. One has

:<math>V(\mathbf{r})=\frac{-g^2}{(2\pi)^3} \int e^{i\mathbf{k \cdot r}}
\frac {4\pi}{k^2+m^2} \;d^3k</math>

where the integral is performed over all possible values of the 3-vector momentum ''k''. In this form, the fraction <math>4\pi/(k^2+m^2)</math> is seen to be the [[propagator]] or [[Green's function]] of the [[Klein-Gordon equation]].

==Feynman amplitude==
[[Image:1pxchg.svg|right|Single particle exchange]]
The Yukawa potential can be derived as the lowest order amplitude of the interaction of a pair of fermions. The [[Yukawa interaction]] couples the fermion field <math>\psi(x)</math> to the meson field <math>\phi(x)</math> with the coupling term

:<math>\mathcal{L}_\mathrm{int}(x) = g\overline{\psi}(x)\phi(x) \psi(x).</math>

The [[scattering amplitude]] for two fermions, one with initial momentum <math>p_1</math> and the other with momentum <math>p_2</math>, exchanging a meson with momentum ''k'', is given by the [[Feynman diagram]] on the right.

The Feynman rules for each vertex associate a factor of ''g'' with the amplitude; since this diagram has two vertices, the total amplitude will have a factor of <math>g^2</math>. The line in the middle, connecting the two fermion lines, represents the exchange of a meson. The Feynman rule for a particle exchange is to use the propagator; the propagator for a massive meson is <math>-4\pi/(k^2+m^2)</math>. Thus, we see that the Feynman amplitude for this graph is nothing more than

:<math>V(\mathbf{k})=-g^2\frac{4\pi}{k^2+m^2}.</math>

From the previous section, this is seen to be the Fourier transform of the Yukawa potential.
{{clr}}

==See also==
*[[Yukawa interaction]]
*[[Screened Poisson equation]]

==References==
===Citations===
{{reflist}}

===Texts===
* [[Hideki Yukawa|H. Yukawa]], ''On the interaction of elementary particles''. (1935) Proc. Phys. Math. Soc. Japan. '''17''' 48

* [[Gerald Edward Brown]] and A. D. Jackson, ''The Nucleon-Nucleon Interaction'', (1976) North-Holland Publishing, Amsterdam ISBN 0-7204-0335-9

[[Category:Particle physics]]
[[Category:Quantum field theory]]
[[Category:Potentials]]