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Path Integrals in Quantum Mechanics$
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Jean Zinn-Justin

Print publication date: 2004

Print ISBN-13: 9780198566748

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198566748.001.0001

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BARRIER PENETRATION: SEMI-CLASSICAL APPROXIMATION

BARRIER PENETRATION: SEMI-CLASSICAL APPROXIMATION

Chapter:
(p.225) 8 BARRIER PENETRATION: SEMI-CLASSICAL APPROXIMATION
Source:
Path Integrals in Quantum Mechanics
Author(s):

Jean Zinn-Justin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566748.003.0008

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.

Keywords:   path integral, barrier penetration, semi-classical approximation, semi-classical limit, degenerate minima, metastable states

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