# Yukawa potential

In particle and atomic physics, a Yukawa potential (also called a screened Coulomb potential) is a potential of the form

V

Yukawa

(
r
)
=

g

2

e

k
m
r

r

,

{\displaystyle V_{\text{Yukawa}}(r)=-g^{2}{\frac {e^{-kmr}}{r}},}

where g is a magnitude scaling constant, i.e. is the amplitude of potential, m is the mass of the particle, r is the radial distance to the particle, and k is another scaling constant, so that 1/km is the range. The potential is monotone increasing in r and it is negative, implying the force is attractive. In the SI system, the unit of the Yukawa potential is (1/m).
The Coulomb potential of electromagnetism is an example of a Yukawa potential with e−kmr equal to 1 everywhere. This can be interpreted as saying 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 gauge coupling constant between those fields. In the case of the nuclear force, the fermions would be a proton and another proton or a neutron.

Contents

1 History
2 Relation to Coulomb potential
3 Fourier transform
4 Feynman amplitude
5 Eigenvalues of Schrödinger equation
7 References

7.1 Citations
7.2 Texts

History
Hideki Yukawa showed in the 1930s that such a potential arises from the exchange of a massive scalar field such as the field of a massive boson. Since the field mediator is massive the corresponding force has a certain range, which is inversely proportional to the mass of the mediator particle

m

{\displaystyle m}

.[1] Because the approximate range of the nuclear force was known, Yukawa’s equation could be used to predict the approximate rest mass of the particle mediating the force field, even before it was discovered. In the case of the nuclear force, this mass was predicted to be about 200 times the mass of the electron, and this was later considered to be a prediction of the existence of the pion, before it was detected in 1947.
Relation to Coulomb potential

Figure 1: A comparison of Yukawa potentials where