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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 AND QUANTIZATION

PATH INTEGRALS AND QUANTIZATION

Chapter:
(p.111) 5 PATH INTEGRALS AND QUANTIZATION
Source:
Path Integrals in Quantum Mechanics
Author(s):

Jean Zinn-Justin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198566748.003.0005

This chapter constructs path integrals for general Hamiltonians with potentials linear in the velocities, like Hamiltonians of particles in a magnetic field. Two examples are considered: a quantum system coupled to a magnetic field, and diffusion as described by the Fokker-Planck equation. In both examples, the Hamiltonian contains products of the position and momentum operators. A quantization problem then arises since these operators do not commute, and the correspondence principle is no longer sufficient to determine the quantum Hamiltonian. The order between quantum operators is determined by additional conditions, such as hermiticity or conservation of probabilities. The calculation of the corresponding path integral then suffers from ambiguities, directly related to this quantization problem. The continuum limit is no longer unique, but depends on the limiting process. This chapter also considers a situation where space has a nontrivial topology, in this case a circle, and shows how this influences the calculation of the path integral.

Keywords:   path integrals, quantization, magnetic field, Fokker-Planck equation, Hamiltonians, diffusion, quantum operators, potentials

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