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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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PATH INTEGRALS IN QUANTUM MECHANICS

PATH INTEGRALS IN QUANTUM MECHANICS

Chapter:
(p.27) 2 PATH INTEGRALS IN QUANTUM MECHANICS
Source:
Path Integrals in Quantum Mechanics
Author(s):

Jean Zinn-Justin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566748.003.0002

This chapter constructs the path integral associated with the statistical operator e -βH in the case of Hamiltonians of the simple form p2/2m + V (q). The path integral corresponding to a harmonic oscillator coupled to an external, time-dependent force is then calculated. This result allows a perturbative evaluation of path integrals with general analytic potentials. The results are applied to the calculation of the partition function tr e-βH using perturbative and semi-classical methods. The integrand for this class of path integrals defines a positive measure on paths. It is thus natural to introduce the corresponding expectation values, called correlation functions. Moments of such a distribution can be generated by a generating functional, and recovered by functional differentiation. These results can be applied to the determination of the spectrum of a class of Hamiltonians in several approximation schemes.

Keywords:   path integrals, statistical operator, correlation functions, partition function, harmonic oscillator, expectation values, Hamiltonians, generating functional

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