The Sampling Theorem and Coherent State Systems in Quantum Mechanics

Details The Sampling Theorem and Coherent State Systems in Quantum Mechanics The Sampling Theorem and Coherent State Systems in Quantum Mechanics

2006-01-10

arxiv.org/abs/quant-ph/0601059v1

Physics

Quantum Physics

20 pages, Latex

Scientific paper

10.1088/0031-8949/74/2/004

The well known Poisson Summation Formula is analysed from the perspective of the coherent state systems associated with the Heisenberg--Weyl group. In particular, it is shown that the Poisson summation formula may be viewed abstractly as a relation between two sets of bases (Zak bases) arising as simultaneous eigenvectors of two commuting unitary operators in which geometric phase plays a key role. The Zak bases are shown to be interpretable as generalised coherent state systems of the Heisenberg--Weyl group and this, in turn, prompts analysis of the sampling theorem (an important and useful consequence of the Poisson Summation Formula) and its extension from a coherent state point of view leading to interesting results on properties of von Neumann and finer lattices based on standard and generalised coherent state systems.

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