Many-body problem
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The quantum many-body problem is a general name for a vast category of physical problems pertaining to deriving the behavior of multi-particle systems using fundamental quantum-mechanical principles.[1]
The goal of many-body physics is to find new principles to describe macroscopic systems, using principles that pertain to microscopic systems.[2]
Terminology
[edit]Many can be anywhere from three to infinity, although three- and four-body systems can be treated by specific means (respectively the Faddeev[3] and Faddeev–Yakubovsky[4] equations) and are thus sometimes separately classified as few-body systems.
Body, in this case, is referring to a particle (electron, nuclei, atom, etc.).[1]
Explanation
[edit]All information about a system can be extracted from its wave function. Solving Schrödinger's equation allows you to find the wave function associated with a system, and therefore determine the properties of it. Schrödinger's equation can only be solved exactly for the hydrogen atom or simpler systems; anything more complex than this can only be solved through approximation methods. This is because when the amount of particles in a system increases, repulsive interactions between electrons occur, and the term in Schrödinger's equation that corresponds to the energy of these repulsions is impossible to solve for.[5]
This arising complexity becomes especially clear by a comparison to classical mechanics. Imagine a single particle that can be described with numbers (take for example a free particle described by its position and velocity vector, resulting in ). In classical mechanics, such particles can simply be described by numbers. The dimension of the classical many-body system scales linearly with the number of particles, .
In quantum mechanics, however, the dimension of the many-body wave function scales exponentially with , much faster than in classical mechanics.[2]
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.[6] Thus, many-body physics most often relies on a set of approximation methods, such as the variational method and perturbation theory, specific to the problem at hand. It ranks among the most computationally intensive fields of science.[5]
Many-body problems play a central role in condensed matter physics.[1]
Examples
[edit]- 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
[edit]- 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
- Matrix product state
- Neural network quantum states
- Numerical renormalization group
Further reading
[edit]- Jenkins, Stephen. "The Many Body Problem and Density Functional Theory".
- 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.
References
[edit]- 1 2 3 Lindsey, Michael (2019). The Quantum Many-Body Problem: Methods and Analysis (Thesis). UC Berkeley.
- 1 2 Coleman, Piers (2015-11-26). Introduction to Many-Body Physics. Cambridge University Press. ISBN 978-0-521-86488-6.
- ↑ Nielsen, E.; Fedorov, D. V.; Jensen, A. S.; Garrido, E. (2001-06-01). "The three-body problem with short-range interactions". Physics Reports. 347 (5): 373–459. Bibcode:2001PhR...347..373N. doi:10.1016/S0370-1573(00)00107-1. ISSN 0370-1573.
- ↑ Lazauskas, Rimantas; Carbonell, Jaume (2020). "Description of Four- and Five-Nucleon Systems by Solving Faddeev-Yakubovsky Equations in Configuration Space". Frontiers in Physics. 7 251. arXiv:2002.05876. Bibcode:2020FrP.....7..251L. doi:10.3389/fphy.2019.00251.
- 1 2 McQuarrie, Donald A.; Simon, John D. (200). Physical chemistry: a molecular approach. Sausalito, Calif: Univ. Science Books. ISBN 978-0-935702-99-6.
{{cite book}}: ISBN / Date incompatibility (help) - ↑ Hochstuhl, David; Bonitz, Michael; Hinz, Christopher (2014). "Time-dependent multiconfiguration methods for the numerical simulation of photoionization processes of many-electron atoms". The European Physical Journal Special Topics. 223 (2): 177–336. Bibcode:2014EPJST.223..177H. doi:10.1140/epjst/e2014-02092-3. S2CID 122869981.
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