On the Relationship Between Complex Potentials and Strings of Projection Operators

Abstract

It is of interest in a variety of contexts, and in particular in the arrival time problem, to consider the quantum state obtained through unitary evolution of an initial state regularly interspersed with periodic projections onto the positive $x$-axis (pulsed measurements). Echanobe, del Campo and Muga have given a compelling but heuristic argument that the state thus obtained is approximately equivalent to the state obtained by evolving in the presence of a certain complex potential of step-function form. In this paper, with the help of the path decomposition expansion of the associated propagators, we give a detailed derivation of this approximate equivalence. The propagator for the complex potential is known so the bulk of the derivation consists of an approximate evaluation of the propagator for the free particle interspersed with periodic position projections. This approximate equivalence may be used to show that to produce significant reflection, the projections must act at time spacing less than 1/E, where E is the energy scale of the initial state.

29 pages, LaTex, 4 figures. Substantial revisions

References in corpus (5)

• Probabilities in Quantum Cosmological Models: A Decoherent Histories Analysis Using a Complex Potential
• Arrival Times, Complex Potentials and Decoherent Histories
• Disclosing hidden information in the quantum Zeno effect: Pulsed measurement of the quantum time of arrival
• Quantum Arrival Time Formula from Decoherent Histories
• Spacetime Coarse Grainings in the Decoherent Histories Approach to Quantum Theory

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• Pitfalls of Path Integrals: Amplitudes for Spacetime Regions and the Quantum Zeno Effect
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• Amplitudes for Spacetime Regions and the Quantum Zeno Effect: Pitfalls of Standard Path Integral Constructions
• A review of the decoherent histories approach to the arrival time problem in quantum theory