Yukawa potential

The potential is monotonically increasing in r and it is negative, implying the force is attractive.

In the SI system, the unit of the Yukawa potential is the inverse meter.

Prior to Hideki Yukawa's 1935 paper,[1] physicists struggled to explain the results of James Chadwick's atomic model, which consisted of positively charged protons and neutrons packed inside of a small nucleus, with a radius on the order of 10−14 meters.

Physicists knew that electromagnetic forces at these lengths would cause these protons to repel each other and for the nucleus to fall apart.

In 1932, Werner Heisenberg proposed a "Platzwechsel" (migration) interaction between the neutrons and protons inside the nucleus, in which neutrons were composite particles of protons and electrons.

When, in 1933 at the Solvay Conference, Heisenberg proposed his interaction, physicists suspected it to be of either two forms: on account of its short-range.

[3] Fermi's neutron-proton interaction was not based on the "migration" of neutrons and protons between each other.

While Fermi's interaction solved the issue of the conservation of linear and angular momentum, Soviet physicists Igor Tamm and Dmitri Ivanenko demonstrated that the force associated with the neutrino and electron emission was not strong enough to bind the protons and neutrons in the nucleus.

[4] In his February 1935 paper, Hideki Yukawa combines both the idea of Heisenberg's short-range force interaction and Fermi's idea of an exchange particle in order to fix the issue of the neutron-proton interaction.

He deduced a potential which includes an exponential decay term (

In the case of QED, this exchange particle was a photon of 0 mass.

In Yukawa's case, the exchange particle had some mass, which was related to the range of interaction (given by

Physicists called this particle the "meson," as its mass was in the middle of the proton and electron.

In fact, we have: Consequently, the equation simplifies to the form of the Coulomb potential where we set the scaling constant to be:[5] A comparison of the long range potential strength for Yukawa and Coulomb is shown in Figure 2.

One has where the integral is performed over all possible values of the 3-vector momenta k. In this form, and setting the scaling factor to one,

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

with the coupling term The scattering amplitude for two fermions, one with initial momentum

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.

The radial Schrödinger equation with Yukawa potential can be solved perturbatively.[6][7][8]: ch.

equal to zero, one obtains the well-known expression for the Schrödinger eigenvalue for the Coulomb potential, and the radial quantum number

is a positive integer or zero as a consequence of the boundary conditions which the wave functions of the Coulomb potential have to satisfy.

In the case of the Yukawa potential the imposition of boundary conditions is more complicated.

The above expansion for the orbital angular momentum or Regge trajectory

, as was argued originally by Langer,[10] the reason being that the singularity is too strong for an unchanged application of the WKB method.

[11] We can calculate the differential cross section between a proton or neutron and the pion by making use of the Yukawa potential.

We use the Born approximation, which tells us that, in a spherically symmetrical potential, we can approximate the outgoing scattered wave function as the sum of incoming plane wave function and a small perturbation: where

: Evaluating the integral gives Energy conservation implies so that Plugging in, we get: We thus get a differential cross section of:[5] Integrating, the total cross section is: The potential outside of an infinitesimally thin, uniform spherical shell with total scaling constant

is also a Yukawa potential, but in general the scaling contstant for the equivalent point source is larger than for the shell.

, then one recovers the shell theorem for the inverse square potential.

A consequence of this is that in modified gravity theories where the graviton has nonzero mass, the weak equivalence principle would be violated and the gravitational acceleration of a body in free fall would depend on its composition.