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Yukawa potential

 

 
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Yukawa potential
 
 
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A Yukawa potential (also called a 'screened Coulomb potential') is a potentialPotential

In physics, a potential may refer to the scalar potential or to the vector potential....
 of the form

Hideki YukawaHideki Yukawa

Hideki Yukawa FRSE was a Japanese theoretical physicist and the first Japanese to win the Nobel prize....
 showed in the 1930s that such a potential arises from the exchange of a massive scalar fieldScalar field (quantum field theory)

In quantum field theory, a scalar field is a quantum field whose quanta are spin-zero particles....
 such as the field of the pionPion

In particle physics, pion is the collective name for three subatomic particles: π0, π+ and π−....
 whose mass is . Since the field mediator is massive the corresponding force has a certain range due to its decay, which range is inversely proportional to the mass. If the mass is zero, then the Yukawa potential becomes equivalent to a Coulomb potential, and the range is said to be infinite.

In the above equation, the potential is negative, denoting that the force is attractive. The constant g is a real number; it is equal to the coupling constantFacts About Coupling constant

In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction....
 between the meson field and the fermionFermion

In particle physics, fermions are particles with half-integer spin....
 field with which it interacts. In the case of nuclear physicsFacts About Nuclear physics

Nuclear physics is the branch of physics concerned with the nucleus of the atom....
, the fermions would be the protonProton

In physics, the proton is a subatomic particle with an electric charge of one positive fundamental unit , a diameter of abo...
 and the neutronNeutron

In physics, the neutron is a subatomic particle with no net electric charge and a mass of 939.573 MeV/c ....
.

Fourier transform

The easiest way to understand that the Yukawa potential is associated with a massive field is by examining its Fourier transformFourier transform

The Fourier transform, named after Joseph Fourier, is a reversible integral transform of one function into another....
. One has

where the integral is performed over all possible values of the 3-vector momentum k. In this form, the fraction is seen to be the propagatorPropagator Overview

In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from ...
 or Green's functionGreen's function

In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to bound...
 of the Klein-Gordon equationKlein-Gordon equation

The Klein-Gordon equation is the relativistic version of the Schrdinger equation, which is used to describe spinless particl...
.

Feynman amplitude

The Yukawa potential can be derived as the lowest order amplitude of the interaction of a pair of fermions. The Yukawa interactionYukawa interaction

In particle physics, Yukawa interaction, named after Hideki Yukawa, is an interaction between a scalar field and a Dirac fi...
 couples the fermion field to the meson field with the coupling term

The scattering amplitudeScattering amplitude

The scattering amplitude describes the amplitude of an outgoing, elementary, spherical wave relative to a plane, incoming wa...
 for two fermions, one with initial momentum and the other with momentum , exchanging a meson with momentum k, is given by the Feynman diagramFeynman diagram

A Feynman diagram is a method for performing calculations in quantum field theory, invented by American physicist Richard Fe...
 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 . 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 . Thus, we see that the Feynman amplitude for this graph is nothing more than

From the previous section, this is clearly seen to be the Fourier transform of the Yukawa pote .