Many-body problem

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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 may be defined as the study of the effects of interaction between bodies on the behaviour of a many-body system, i.e. a closed system which does not contain just a few bodies in action, such as the collisions discussed in classical mechanics. Due to the number of particles/bodies contained in such a system, the number of degrees of freedom increases rapidly, and it becomes difficult to describe the mechanics of the system by using a small system of equations.

The many-body problem is usually posed in quantum mechanics as the question of solving for more complex problems than the hydrogen atom. Depending on the complexity of the molecule, different models are used. For example, a many-body problem posed for a polymer molecule would be different from a single polyatomic molecule. The former might include some statistical parameters, whereas the latter would likely exclude these.

Contents 1 Approaches 2 Examples 3 Quotes 4 See also 5 Further reading

[edit] Approaches

In some many-body problems, the solutions are chaotic. Sometimes, many-body problems can be simplified by canonicalization.

Another approach to solve many-body problems is to simply ignore some interactions within the system. This allows the many-body problem to be reduced to a simpler problem, often a set of smaller, independent and easier to solve problems.

[edit] Examples

Solving the molecular Hamiltonian for a particular chemical system is an example of a many body problem.

Among the real world many-body systems, only a few have been analytically (exactly) solved for their Schrödinger wave functions:

1. H₂⁺ (hydrogen molecular ion, two nuclei and one electron): Its wave function was first solved by O. Burrau in 1927 based on a semi-classical strategy (O. Burrau, Danske Vidensk. Selskab. Math.-fys. Meddel., 7(14), 1 (1927)). More extensive and general quantization solutions have been given by other scholars, notably Bates et al. (D. R. Bates, K. Ledsham, A. L. Stewart, Wave Functions of the Hydrogen Molecular Ion, Phil. Trans. Royal Soc. London. Series A, Math. Phys. Sci., 246, 215-240 (1953)[1]), and Ezra et al. (G. S. Ezra, C. C. Martens, L. E. Fried, Semiclassical quantization of polyatomic molecules: some recent developments, J. Phys. Chem. 91, 3721-3730 (1987)[2]).

2. He (helium atom, one nucleus and two electrons): Its wave function was first solved by Feng Xu in 1989 based on a semi-classical strategy similar to that applied to H₂⁺ (F. Xu, Application of a H₂⁺-like model to helium atom. Solution of the wave equation at the ground state. Z. Phys. D 13, 279-280 (1989)[3]). Other exact solutions were also provided in later publications, such as that of Zhang and Deng (R. Zhang and C. Deng, Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems. Phys. Rev. A 47, 71 - 77 (1993)[4]).

3. H^- (hydrogen anion or hydride, one nucleus and two electrons): Its wave function was solved in several publications, such as that of Zhang and Deng (R. Zhang and C. Deng, Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems. Phys. Rev. A 47, 71 - 77 (1993)[5]).

[edit] 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

[edit] See also

• Hartree-Fock approximation

[edit] Further reading

• Stephen Jenkins: The Many Body Problem and Density Functional Theory
• D. J. Thouless: The quantum mechanics of many-body systems. New York, Academic Press, 1972, ISBN 0-12-691560-1.