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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Degenerate Classical Minima And Instantons

Degenerate Classical Minima And Instantons

Chapter:
(p.980) 41 DEGENERATE CLASSICAL MINIMA AND INSTANTONS
Source:
Quantum Field Theory and Critical Phenomena
Author(s):

JEAN ZINN-JUSTIN

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198509233.003.0041

This chapter considers a somewhat different situation in which instantons play a role: classical actions with degenerate isolated minima. The simplest examples are provided by simple quantum mechanics when potentials have degenerate minima. Classically the state of minimum energy corresponds to a particle sitting at any of the minima of the potential. The position of the particle breaks (spontaneously) the symmetry of the system. In quantum mechanics on the contrary, we expect the modulus of the ground state wave function to be large near all minima of the potential, as a consequence of barrier penetration effects. The chapter discusses this phenomenon on two typical examples: the double-well potential and the periodic cosine potential. Section 41.3 relates instantons to Arrhenius law in the context of stochastic dynamics.

Keywords:   double-well potential, instantons, degenerate minima, quantum mechanics, periodic cosine potential, Arrhenius law

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