*Jean Zinn-Justin*

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.001.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

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 ...
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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.Less

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.

*Jean Zinn-Justin*

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0008
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

A classical particle is always reflected by a potential barrier if its energy is lower than the potential. In contrast, a quantum particle has a non-vanishing probability to tunnel through a barrier, ...
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A classical particle is always reflected by a potential barrier if its energy is lower than the potential. In contrast, a quantum particle has a non-vanishing probability to tunnel through a barrier, a property known as barrier penetration. This chapter examines various physical manifestations of barrier penetration in the semi-classical approximation using the path integral formalism. Two specific problems are discussed: the splitting between two classically degenerate energy levels corresponding to two symmetric minima of a potential are evaluated in the semi-classical limit; in the same limit, the decay rate, and thus the lifetime, of metastable states is also calculated. The chapter focuses first on a family of quantum systems where tunneling plays a role: the Hamiltonian has a discrete space symmetry, but the potential has minima at points that are not group invariant. The positions of the degenerate minima are then related by symmetry group transformations.Less

A classical particle is always reflected by a potential barrier if its energy is lower than the potential. In contrast, a quantum particle has a non-vanishing probability to tunnel through a barrier, a property known as barrier penetration. This chapter examines various physical manifestations of barrier penetration in the semi-classical approximation using the path integral formalism. Two specific problems are discussed: the splitting between two classically degenerate energy levels corresponding to two symmetric minima of a potential are evaluated in the semi-classical limit; in the same limit, the decay rate, and thus the lifetime, of metastable states is also calculated. The chapter focuses first on a family of quantum systems where tunneling plays a role: the Hamiltonian has a discrete space symmetry, but the potential has minima at points that are not group invariant. The positions of the degenerate minima are then related by symmetry group transformations.