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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 PHASE SPACE

PATH INTEGRALS IN PHASE SPACE

Chapter:
(p.279) 10 PATH INTEGRALS IN PHASE SPACE
Source:
Path Integrals in Quantum Mechanics
Author(s):

Jean Zinn-Justin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566748.003.0010

This chapter contains a few additional results such as a definition of path integrals over phase space trajectories, and the problems generated by the quantization of lagrangians with potentials quadratic in the velocities. The important example of Hamiltonians quadratic in the momentum variables is first examined. In the simplest situations discussed in previous chapters, after an explicit integration over momenta p(t) one recovers the usual path integral. More general Hamiltonians are often met, for example, in the quantization of the motion on riemannian manifolds. The analysis is illustrated with the quantization of free motion on a sphere (or hypersphere) S N-1. A few relevant elements of classical mechanics are considered first.

Keywords:   path integrals, phase space, classical mechanics, quantization, free motion, sphere, lagrangians, Hamiltonians, momentum variables

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