Faddeev equations
Encyclopedia
The Faddeev equations, named after their inventor Ludvig Faddeev
, are equation
s that describe, at once, all the possible exchanges/interaction
s in a system of three particles
in a fully quantum mechanical
formulation. They can be solved iteratively.
In general, Faddeev equations need as input a potential
that describes the interaction between two individual particles. It is also possible to introduce a term in the equation in order to take also three-body force
s into account.
The Faddeev equations are the most often used non-perturbative
formulations of the quantum-mechanical three-body problem.
Unlike the three body problem in classical mechanics
, the quantum three body problem is uniformly soluble.
In nuclear physics
, the off the energy shell
nucleon-nucleon interaction has been studied by analyzing (n,2n) and (p,2p) reactions on deuterium
targets, using the Faddeev Equations. The nucleon-nucleon interaction is expanded (approximated) as a series
of separable potentials. The Coulomb
interaction between two protons is a special problem, in that its expansion in separable potentials does not converge, but this is handled by matching the Faddeev solutions to long range coulomb solutions, instead of to plane waves.
Separable potentials are interactions that do not preserve a particle's location. Ordinary local potential
s can be expressed as sums of separable potentials. The physical nucleon-nucleon interaction, which involves exchange of meson
s, is not expected to be either local or separable.
Ludvig Faddeev
-References:...
, are equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
s that describe, at once, all the possible exchanges/interaction
Interaction
Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect...
s in a system of three particles
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
in a fully quantum mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
formulation. They can be solved iteratively.
In general, Faddeev equations need as input a potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
that describes the interaction between two individual particles. It is also possible to introduce a term in the equation in order to take also three-body force
Three-body force
A three-body force is a force that does not exist in a system of two objects but appears in a three-body system. In general, if the behaviour of a system of more than two objects cannot be described by the two-body interactions between all possible pairs, as a first approximation, the deviation is...
s into account.
The Faddeev equations are the most often used non-perturbative
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
formulations of the quantum-mechanical three-body problem.
Unlike the three body problem in classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, the quantum three body problem is uniformly soluble.
In nuclear physics
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
, the off the energy shell
On shell and off shell
In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell....
nucleon-nucleon interaction has been studied by analyzing (n,2n) and (p,2p) reactions on deuterium
Deuterium
Deuterium, also called heavy hydrogen, is one of two stable isotopes of hydrogen. It has a natural abundance in Earth's oceans of about one atom in of hydrogen . Deuterium accounts for approximately 0.0156% of all naturally occurring hydrogen in Earth's oceans, while the most common isotope ...
targets, using the Faddeev Equations. The nucleon-nucleon interaction is expanded (approximated) as a series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
of separable potentials. The Coulomb
Coulomb's law
Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
interaction between two protons is a special problem, in that its expansion in separable potentials does not converge, but this is handled by matching the Faddeev solutions to long range coulomb solutions, instead of to plane waves.
Separable potentials are interactions that do not preserve a particle's location. Ordinary local potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
s can be expressed as sums of separable potentials. The physical nucleon-nucleon interaction, which involves exchange of meson
Meson
In particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...
s, is not expected to be either local or separable.