The Faddeev equations, named after their inventor

Ludvig Faddeev Ludvig Dmitrievich Faddeev (also ''Ludwig Dmitriyevich''; russian: Лю́двиг Дми́триевич Фадде́ев; 23 March 1934 – 26 February 2017) was a Soviet and Russian mathematical physicist. He is known for the discovery of the ...

, are equations that describe, at once, all the possible exchanges/

interaction Interaction is action that occurs between two or more objects, with broad use in philosophy and the sciences. It may refer to: Science * Interaction hypothesis, a theory of second language acquisition * Interaction (statistics) * Interactions o ...

s in a system of three

particles 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 s ...

in a fully

quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...

formulation. They can be solved iteratively. In general, Faddeev equations need as input 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 ...

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 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 possib ...

s into account. The Faddeev equations are the most often used non-

perturbative In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for whi ...

formulations of the quantum-mechanical three-body problem. Unlike the three body problem in

classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...

, the quantum three body problem is uniformly soluble. In

nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...

, the off the energy shell nucleon-nucleon interaction has been studied by analyzing (n,2n) and (p,2p) reactions on

deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one ...

targets, using the Faddeev Equations. The nucleon-nucleon interaction is expanded (approximated) as a

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...

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 In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, t ...

. Separable potentials are interactions that do not preserve a particle's location. Ordinary local

s can be expressed as sums of separable potentials. The physical nucleon-nucleon interaction, which involves exchange of

meson In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, ...

s, is not expected to be either local or separable.

References

* L.D. Faddeev, S.P. Merkuriev, Quantum Scattering Theory for Several Particle Systems, Springer, August 31, 1993, . {{DEFAULTSORT:Faddeev Equations Quantum mechanics Nuclear physics Equations