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

*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds. This mathematical formulation arises from the fact that, in physics, the scalar potential is irrotational, and thus has a vanishing Laplacian ? the very definition of a harmonic function....
 of the form

Hideki Yukawa
Hideki Yukawa

n? , was a Japanese theoretical physicist and the first Japanese Nobel prize....
 showed in the 1930s that such a potential arises from the exchange of a massive scalar field such as the field of the pion
Pion

In particle physics, a pion is any of three subatomic particles: , and . Pions are the lightest mesons and play an important role in explaining low-energy properties of the strong nuclear force....
 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.






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

*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds. This mathematical formulation arises from the fact that, in physics, the scalar potential is irrotational, and thus has a vanishing Laplacian ? the very definition of a harmonic function....
 of the form

Hideki Yukawa
Hideki Yukawa

n? , was a Japanese theoretical physicist and the first Japanese Nobel prize....
 showed in the 1930s that such a potential arises from the exchange of a massive scalar field such as the field of the pion
Pion

In particle physics, a pion is any of three subatomic particles: , and . Pions are the lightest mesons and play an important role in explaining low-energy properties of the strong nuclear force....
 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 constant
Coupling constant

In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian mechanics of a system can be separated into a kinetic part and an interaction part....
 between the meson field and the fermion
Fermion

In particle physics, fermions are subatomic particle which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle....
 field with which it interacts. In the case of the nuclear force
Nuclear force

The nuclear force is the force between two or more nucleons. It is responsible for binding of protons and neutrons into Atomic nucleus. To a large extent, this force can be understood in terms of the exchange of virtual light mesons, such as the pions....
, the fermions would be the proton
Proton

The proton is a subatomic particle with an electric charge of +1 elementary charge. It is found in the nucleus of each atom but is also stable by itself and has a second identity as the hydrogen ion, H+....
 and another proton or the neutron
Neutron

The neutron is a subatomic particle with no net electric charge and a mass slightly larger than that of a proton.Neutrons are usually found in atomic nucleus....
.

Fourier transform

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

In mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions....
. 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 propagator
Propagator

In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum....
 or Green's function
Green's function

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

The Klein?Gordon equation is a special relativity version of the Schr?dinger equation.It is the equation of motion of a quantum field theory, a field whose quanta are spinless particles....
.

Feynman amplitude

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

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

The scattering amplitude
Scattering amplitude

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

In quantum field theory a Feynman diagram is an intuitive graphical representation of a contribution to the transition amplitude or correlation function of a quantum mechanical or statistical field theory....
 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 seen to be the Fourier transform of the Yukawa potential.