HOME

TheInfoList



OR:

The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that
quantum mechanics 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, ...
has to be used to provide an accurate description of the system. ''Many'' can be anywhere from three to infinity (in the case of a practically infinite,
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size ...
or periodic system, such as a
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macr ...
), although three- and four-body systems can be treated by specific means (respectively the Faddeev and Faddeev–Yakubovsky equations) and are thus sometimes separately classified as
few-body systems In mechanics, a few-body system consists of a small number of well-defined structures or point particles. Quantum mechanics In quantum mechanics, examples of few-body systems include light nuclear systems (that is, few-nucleon bound and scatteri ...
. In general terms, while the underlying physical laws that govern the motion of each individual particle may (or may not) be simple, the study of the collection of particles can be extremely complex. In such a quantum system, the repeated interactions between particles create quantum correlations, or entanglement. As a consequence, the
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
of the system is a complicated object holding a large amount of
information Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random, ...
, which usually makes exact or analytical calculations impractical or even impossible. This becomes especially clear by a comparison to classical mechanics. Imagine a single particle that can be described with k numbers (take for example a free particle described by its position and velocity vector, resulting in k=6). In classical mechanics, n such particles can simply be described by k\cdot n numbers. The dimension of the classical many-body system scales linearly with the number of particles n . In quantum mechanics, however, the many-body-system is in general in a superposition of combinations of single particle states - all the k^n different combinations have to be accounted for. The dimension of the quantum many body system therefore scales exponentially with n , much faster than in classical mechanics. Because the required numerical expense grows so quickly, simulating the dynamics of more than three quantum-mechanical particles is already infeasible for many physical systems. Thus, many-body theoretical physics most often relies on a set of
approximations An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix '' ...
specific to the problem at hand, and ranks among the most computationally intensive fields of science. In many cases, emergent phenomena may arise which bear little resemblance to the underlying elementary laws. Many-body problems play a central role in
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 ...
.


Examples

*
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 ...
(
solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the ...
,
nanoscience The nanoscopic scale (or nanoscale) usually refers to structures with a length scale applicable to nanotechnology, usually cited as 1–100 nanometers (nm). A nanometer is a billionth of a meter. The nanoscopic scale is (roughly speaking) a l ...
,
superconductivity Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
) *
Bose–Einstein condensation Bose–Einstein may refer to: * Bose–Einstein condensate ** Bose–Einstein condensation (network theory) * Bose–Einstein correlations * Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describ ...
and
Superfluids Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in tw ...
*
Quantum chemistry Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
(
computational chemistry Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of mol ...
,
molecular physics Molecular physics is the study of the physical properties of molecules and molecular dynamics. The field overlaps significantly with physical chemistry, chemical physics, and quantum chemistry. It is often considered as a sub-field of atomic, m ...
) *
Atomic physics Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
*
Molecular physics Molecular physics is the study of the physical properties of molecules and molecular dynamics. The field overlaps significantly with physical chemistry, chemical physics, and quantum chemistry. It is often considered as a sub-field of atomic, m ...
*
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 t ...
(
Nuclear structure Understanding the structure of the atomic nucleus is one of the central challenges in nuclear physics. Models The liquid drop model The liquid drop model is one of the first models of nuclear structure, proposed by Carl Friedrich von Weizsäcke ...
,
nuclear reactions In nuclear physics and nuclear chemistry, a nuclear reaction is a process in which two nuclei, or a nucleus and an external subatomic particle, collide to produce one or more new nuclides. Thus, a nuclear reaction must cause a transformation ...
,
nuclear matter Nuclear matter is an idealized system of interacting nucleons (protons and neutrons) that exists in several phases of exotic matter that, as of yet, are not fully established. It is ''not'' matter in an atomic nucleus, but a hypothetical su ...
) * Quantum chromodynamics (
Lattice QCD Lattice QCD is a well-established non- perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the ...
,
hadron In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the e ...
spectroscopy,
QCD matter Quark matter or QCD matter ( quantum chromodynamic) refers to any of a number of hypothetical phases of matter whose degrees of freedom include quarks and gluons, of which the prominent example is quark-gluon plasma. Several series of conference ...
,
quark–gluon plasma Quark–gluon plasma (QGP) or quark soup is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasma'' signals that free color charges are allowed. In a ...
)


Approaches

* Mean-field theory and extensions (e.g. Hartree–Fock,
Random phase approximation The random phase approximation (RPA) is an approximation method in condensed matter physics and in nuclear physics. It was first introduced by David Bohm and David Pines as an important result in a series of seminal papers of 1952 and 1953. For deca ...
) * Dynamical mean field theory * Many-body
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
and
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differentia ...
-based methods *
Configuration interaction Configuration interaction (CI) is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematica ...
*
Coupled cluster Coupled cluster (CC) is a numerical technique used for describing many-body systems. Its most common use is as one of several post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry, but it is also used in ...
* Various Monte-Carlo approaches *
Density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
*
Lattice gauge theory In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum ele ...
* Matrix product state * Neural network quantum states * Numerical renormalization group


Further reading

* * * * *


References

Quantum mechanics Computational physics {{quantum-stub