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Path Integrals in Quantum Mechanics

Online ISBN:
9780191717994
Print ISBN:
9780198566748
Publisher:
Oxford University Press
Book

Path Integrals in Quantum Mechanics

Jean Zinn-Justin
Jean Zinn-Justin
Head of Dapnia/DSM/CEA-Saclay, France
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Published online:
1 January 2010
Published in print:
18 November 2004
Online ISBN:
9780191717994
Print ISBN:
9780198566748
Publisher:
Oxford University Press

Abstract

Path integrals are mathematical objects that can be considered as generalizations to an infinite number of variables, represented by paths, of usual integrals. They share the algebraic properties of usual integrals, but have new properties from the viewpoint of analysis. They are powerful tools for the study of quantum mechanics, since they emphasize very explicitly the correspondence between classical and quantum mechanics. Physical quantities are expressed as averages over all possible paths but, in the semi-classical limit, the leading contributions come from paths close to classical paths. Thus, path integrals lead to an intuitive understanding of physical quantities in the semi-classical limit, as well as simple calculations of such quantities. This observation can be illustrated with scattering processes, spectral properties, or barrier penetration effects. Even though the formulation of quantum mechanics based on path integrals seems mathematically more complicated than the usual formulation based on partial differential equations, the path integral formulation is well adapted to systems with many degrees of freedom, where a formalism of Schrödinger type is much less useful. It allows simple construction of a many-body theory both for bosons and fermions.

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