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Phase Space Methods for Degenerate Quantum Gases$
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Bryan J. Dalton, John Jeffers, and Stephen M. Barnett

Print publication date: 2014

Print ISBN-13: 9780199562749

Published to Oxford Scholarship Online: April 2015

DOI: 10.1093/acprof:oso/9780199562749.001.0001

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States and Operators

States and Operators

(p.8) 2 States and Operators
Phase Space Methods for Degenerate Quantum Gases

Bryan J. Dalton

John Jeffers

Stephen M. Barnett

Oxford University Press

This chapter sets out the second-quantisation treatment for systems of identical bosons or fermions. Key features such as the symmetrisation principle and super-selection rules are presented. Orthogonality and completeness requirements for mode functions are set out, and Fock states are defined in terms of mode occupancy, leading to the definitions of mode annihilation and creation operators and the determination of their commutation (bosons) or anticommutation (fermions) rules. Important two-mode states such as binomial states and relative-phase eigenstates are considered. Field creation and annihilation operators are defined and related to multi-particle position eigenstates. The Hamiltonian is expressed in terms of both mode operators and field operators, state dynamics being treated via Liouville–von Neumann, master or Matsubara equations for the density operator. Normal ordering is introduced and applied to expressions for the vacuum state projector. Multi-particle position probabilities are considered and related to normally ordered quantum correlation functions.

Keywords:   identical-particle symmetrisation principle, second quantisation, Fock state, mode orthogonality and completeness, super-selection rule, mode and field operators, commutation and anticommutation rules, master equation, normal ordering, quantum correlation function

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