Few-body systems
Encyclopedia
In quantum mechanics
and classical mechanics
, a few-body system consists of a small number of well-defined structures or point particles.
In quantum mechanics, examples of few-body systems include light nuclear systems (that is, few-nucleon bound
and scattering
states), small molecule
s, light atoms (such as helium
in an external electric field
), atomic collisions, and quantum dot
s. A fundamental difficulty in describing few-body systems is that the Schrödinger equation
and the classical equations of motion are not analytically solvable for more than two mutually interacting particles even when the underlying forces are precisely known. This is known as the few-body problem. For some three-body systems an exact solution can be obtained iteratively through the Faddeev equations
. It can be shown that under certain conditions Faddeev equations should lead to Efimov effect. Some special cases of three-body systems are amenable to analytical solutions (or nearly so) - by special treatments - such as the Hydrogen molecular ion whose eigenenergies can be given in terms of a generalized Lambert W function or the Helium atom
which has been solved very precisely using basis sets of Hylleraas or Frankowski-Pekeris functions (see references of the work of G.W.F. Drake and J.D. Morgan III in Helium atom
section).
In many cases theory has to resort to approximations to treat few-body systems. These approximations have to be tested by detailed experimental data. Atomic collisions are particularly suitable for such tests. The fundamental force underlying atomic systems, the electromagnetic force, is essentially understood. Therefore, any discrepancy found between experiment and theory can be directly related to the description of few-body effects. In nuclear system, in contrast, the underlying force is much less understood. Furthermore, in atomic collisions the number of particles can be kept small enough so that complete kinematic information about every single particle in the system can be obtained experimentally. In systems with large particle numbers, in contrast, usually only statistically averaged or collective quantities about the system can be measured.
In classical mechanics, the few-body problem is a subset of the N-body problem
.
One notable journal covering this field is Few-body Systems.
One notable group working in this field is Sector Few-Body Systems at Bogolyubov Theoretical Physics Laboratory of Joint Institute of Nuclear Research in Russia.
Few Body Topical Group at American Physical Society
.
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...
and 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...
, a few-body system consists of a small number of well-defined structures or point particles.
In quantum mechanics, examples of few-body systems include light nuclear systems (that is, few-nucleon bound
Bound state
In physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...
and scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
states), small molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s, light atoms (such as helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...
in an external electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
), atomic collisions, and quantum dot
Quantum dot
A quantum dot is a portion of matter whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei...
s. A fundamental difficulty in describing few-body systems is that the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
and the classical equations of motion are not analytically solvable for more than two mutually interacting particles even when the underlying forces are precisely known. This is known as the few-body problem. For some three-body systems an exact solution can be obtained iteratively through the Faddeev equations
Faddeev equations
The Faddeev equations, named after their inventor Ludvig Faddeev, are equations that describe, at once, all the possible exchanges/interactions in a system of three particles in a fully quantum mechanical formulation. They can be solved iteratively....
. It can be shown that under certain conditions Faddeev equations should lead to Efimov effect. Some special cases of three-body systems are amenable to analytical solutions (or nearly so) - by special treatments - such as the Hydrogen molecular ion whose eigenenergies can be given in terms of a generalized Lambert W function or the Helium atom
Helium atom
Helium is an element and the next simplest atom to solve after the hydrogen atom. Helium is composed of two electrons in orbit around a nucleus containing two protons along with either one or two neutrons, depending on the isotope. The hydrogen atom is used extensively to aid in solving the helium...
which has been solved very precisely using basis sets of Hylleraas or Frankowski-Pekeris functions (see references of the work of G.W.F. Drake and J.D. Morgan III in Helium atom
Helium atom
Helium is an element and the next simplest atom to solve after the hydrogen atom. Helium is composed of two electrons in orbit around a nucleus containing two protons along with either one or two neutrons, depending on the isotope. The hydrogen atom is used extensively to aid in solving the helium...
section).
In many cases theory has to resort to approximations to treat few-body systems. These approximations have to be tested by detailed experimental data. Atomic collisions are particularly suitable for such tests. The fundamental force underlying atomic systems, the electromagnetic force, is essentially understood. Therefore, any discrepancy found between experiment and theory can be directly related to the description of few-body effects. In nuclear system, in contrast, the underlying force is much less understood. Furthermore, in atomic collisions the number of particles can be kept small enough so that complete kinematic information about every single particle in the system can be obtained experimentally. In systems with large particle numbers, in contrast, usually only statistically averaged or collective quantities about the system can be measured.
In classical mechanics, the few-body problem is a subset of the N-body problem
N-body problem
The n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...
.
One notable journal covering this field is Few-body Systems.
One notable group working in this field is Sector Few-Body Systems at Bogolyubov Theoretical Physics Laboratory of Joint Institute of Nuclear Research in Russia.
Few Body Topical Group at American Physical Society
American Physical Society
The American Physical Society is the world's second largest organization of physicists, behind the Deutsche Physikalische Gesellschaft. The Society publishes more than a dozen scientific journals, including the world renowned Physical Review and Physical Review Letters, and organizes more than 20...
.