On the Relationship Between Complex Potentials and Strings of Projection Operators

TL;DR

This paper rigorously derives the approximate equivalence between periodically projected quantum states and evolution under a complex potential, clarifying conditions for significant reflection in pulsed measurement scenarios.

Contribution

It provides a detailed derivation of the equivalence using path decomposition expansion, clarifying the conditions for reflection in pulsed quantum measurements.

Findings

Proves the approximate equivalence between pulsed measurements and complex potential evolution.

Shows that projections must act at intervals less than 1/E to produce significant reflection.

Uses path decomposition expansion to evaluate propagators with periodic projections.

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.