Many-body problem

Quantum mechanics
$\Delta x\, \Delta p \ge \frac{\hbar}{2}$ Uncertainty principle
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This article is about the many-body problem in quantum mechanics. For the n-body problem in classical mechanics, see n-body problem.

The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of a large number of interacting particles. Microscopic here implies that quantum mechanics has to be used to provide an accurate description of the system. A large number can be anywhere from 3 to infinity (in the case of a practically infinite, homogeneous or periodic system, such as a crystal), 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 such a quantum system, the repeated interactions between particles create quantum correlations, or entanglement. As a consequence, the wave function of the system is a complicated object holding a large amount of information, which usually makes exact and/or analytical calculations impractical. Thus, many-body theoretical physics most often relies on a set of approximations specific to the problem at hand, and ranks among the most computationally intensive fields of science.

Examples

• Condensed matter physics (solid-state physics, nanoscience, superconductivity)
• Bose-Einstein condensation and Superfluids
• Quantum chemistry (computational chemistry, molecular physics)
• Atomic physics
• Molecular physics
• Nuclear physics (Nuclear structure, nuclear reactions, nuclear matter)
• Quantum chromodynamics (Lattice QCD, hadron spectroscopy, QCD matter, quark-gluon plasma)

Approaches

• Mean-field theory and extensions (e.g. Hartree-Fock, Random phase approximation)
• Dynamical Mean Field Theory
• Many-body perturbation theory and Green's function-based methods
• Configuration interaction
• Coupled cluster
• Various Monte-Carlo approaches
• Density functional theory
• Lattice gauge theory

Quotes

"It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the many-body problem."

—Max Born, 1960

Further reading

• Jenkins, Stephen. "The Many Body Problem and Density Functional Theory". http://newton.ex.ac.uk/research/qsystems/people/jenkins/mbody/mbody3.html. 
• Thouless, D. J. (1972). The quantum mechanics of many-body systems. New York: Academic Press. ISBN 0-12-691560-1. 
• Fetter, A. L.; Walecka, J. D. (2003). Quantum Theory of Many-Particle Systems. New York: Dover. ISBN 0-486-42827-3. 
• Nozières, P. (1997). Theory of Interacting Fermi Systems. Addison-Wesley. ISBN 0-201-32824-0. 
• Mattuck, R. D. (1976). A guide to Feynman diagrams in the many-body problem. New York: McGraw-Hill. ISBN 0-07-040954-4.