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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]

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]

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 $k$ 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 dimension of the many-body wave function scales exponentially with $n$, 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]

• 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)

• 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