In
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
,
atomic and
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
, a Yukawa potential (also called a screened
Coulomb potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
) is a
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
named after the Japanese physicist
Hideki Yukawa. The potential is of the form:
:
where is a magnitude scaling constant, i.e. is the amplitude of potential, is the mass of the particle, is the radial distance to the particle, and is another scaling constant, so that
is the approximate range. The potential is
monotonically increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
in and it is negative,
implying the force is attractive. In the SI system, the unit of the Yukawa potential is (1/meters).
The
Coulomb potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
of
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
is an example of a Yukawa potential with the
factor equal to 1, everywhere. This can be interpreted as saying that the
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
mass is equal to 0. The photon is the force-carrier between interacting, charged particles.
In interactions between a
meson field and a
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
field, the constant is equal to the
gauge coupling constant between those fields. In the case of the
nuclear force, the fermions would be a
proton
A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
and another proton or a
neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the atomic nucleus, nuclei of atoms. Since protons and ...
.
History
Prior to
Hideki Yukawa's 1935 paper, 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. Thus came the motivation for further explaining the interactions between elementary particles. In 1932,
Werner Heisenberg
Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
proposed a "Platzwechsel" (migration) interaction between the neutrons and protons inside the nucleus, in which neutrons were composite particles of protons and electrons. These composite neutrons would emit electrons, creating an attractive force with the protons, and then turn into protons themselves. When, in 1933 at the
Solvay Conference
The Solvay Conferences (french: Conseils Solvay) have been devoted to outstanding preeminent open problems in both physics and chemistry. They began with the historic invitation-only 1911 Solvay Conference on Physics, considered a turning point i ...
, Heisenberg proposed his interaction, physicists suspected it to be of either two forms:
:
on account of its short-range.
However, there were many issues with his theory. Namely, it is impossible for an electron of spin and a proton of spin to add up to the neutron spin of . The way Heisenberg treated this issue would go on to form the ideas of
isospin.
Heisenberg's idea of an exchange interaction (rather than a Coulombic force) between particles inside the nucleus led Fermi to formulate his ideas on
beta-decay in 1934.
Fermi's neutron-proton interaction was not based on the "migration" of neutron and protons between each other. Instead, Fermi proposed the emission and absorption of two light particles: the neutrino and electron, rather than just the electron (as in Heisenberg's theory). While
Fermi's interaction
In particle physics, Fermi's interaction (also the Fermi theory of beta decay or the Fermi four-fermion interaction) is an explanation of the beta decay, proposed by Enrico Fermi in 1933. The theory posits four fermions directly interactin ...
solved the issue of the conservation of linear and angular momentum, Soviet physicists
Igor Tamm and
Dmitri Ivanenko
Dmitri Dmitrievich Ivanenko (russian: Дми́трий Дми́триевич Иване́нко; July 29, 1904 – December 30, 1994) was a Ukrainian theoretical physicist who made great contributions to the physical science of the twentieth cen ...
demonstrated that the force associated with the neutrino and electron emission was not strong enough to bind the protons and neutrons in the nucleus.
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 (
) and an electromagnetic term (
). In analogy to
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, Yukawa knew that the potential and its corresponding field must be a result of an exchange particle. In the case of
QED, this exchange particle was a
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
of 0 mass. In Yukawa's case, the exchange particle had some mass, which was related to the range of interaction (given by
). Since the range of the nuclear force was known, Yukawa used his equation to predict the mass of the mediating particle as about 200 times the mass of the electron. Physicists called this particle the "
meson," as its mass was in the middle of the proton and electron. Yukawa's meson was found in 1947, and came to be known as the
pion.
Relation to Coulomb potential
If the particle has no mass (i.e., ), then the Yukawa potential reduces to a Coulomb potential, and the range is said to be infinite. In fact, we have:
:
Consequently, the equation
:
simplifies to the form of the Coulomb potential
:
where we set the scaling constant to be:
:
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 is non-zero for any large .
Fourier transform
The easiest way to understand that the Yukawa potential is associated with a massive field is by examining its
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
. One has
:
where the integral is performed over all possible values of the 3-vector momenta . In this form, and setting the scaling factor to one,
, the fraction
is seen to be the
propagator
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. ...
or
Green's function of the
Klein–Gordon equation
The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant ...
.
Feynman amplitude
The Yukawa potential can be derived as the lowest order amplitude of the interaction of a pair of fermions. The
Yukawa interaction
In particle physics, Yukawa's interaction or Yukawa coupling, named after Hideki Yukawa, is an interaction between particles according to the Yukawa potential. Specifically, it is a scalar field (or pseudoscalar field) and a Dirac field of the ...
couples the fermion field
to the meson field
with the coupling term
:
The
scattering amplitude
In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
on the right.
The Feynman rules for each vertex associate a factor of 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.
Eigenvalues of Schrödinger equation
The radial Schrödinger equation with Yukawa potential can be solved perturbatively.
[ Using the radial Schrödinger equation in the form
:
and the Yukawa potential in the power-expanded form
:
and setting , one obtains for the angular momentum the expression
:
for , where
:
Setting all coefficients except 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. Thus in the Yukawa case is only an approximation and the parameter that replaces the integer is really an asymptotic expansion like that above with first approximation the integer value of the corresponding Coulomb case.
The above expansion for the orbital angular momentum or ]Regge trajectory
Regge may refer to
* Tullio Regge (1931-2014), Italian physicist, developer of Regge calculus and Regge theory
* Regge calculus, formalism for producing simplicial approximations of spacetimes
* Regge theory, study of the analytic properties of sc ...
can be reversed to obtain the energy eigenvalues or equivalently . One obtains:
:
The above asymptotic expansion of the angular momentum in descending powers of can also be derived with the WKB method. In that case, however, as in the case of the Coulomb potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
the expression in the centrifugal term of the Schrödinger equation has to be replaced by , as was argued originally by Langer, the reason being that the singularity is too strong for an unchanged application of the WKB method. That this reasoning is correct follows from the WKB derivation of the correct result in the Coulomb case (with the Langer correction), and even of the above expansion in the Yukawa case with higher order WKB approximations.
Cross section
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:
: