Time of arrival operator on the circle. (Variations on two Mielnik's works)

TL;DR

This paper constructs a time-of-arrival operator for a particle on a circle using WWSC quantization, analyzes its properties, and proposes POV measures to describe quantum arrival times, addressing the quantum Zeno effect.

Contribution

It introduces a self-adjoint time-of-arrival operator on the circle and develops POV measures to model arrival times avoiding the quantum Zeno effect.

Findings

The operator is self-adjoint but may not represent a physical observable.

POV measures for arrival times are constructed for the waiting screen scenario.

A method to circumvent the quantum Zeno effect in arrival time measurements is proposed.

Abstract

Using the orthodox Weyl -- Wigner -- Stratonovich -- Cohen (WWSC) quantization rule we construct a time -- of -- arrival operator for a free particle on the circle. It is shown that this operator is self -- adjoint but the careful analysis of its properties suggests that it cannot represent a `physical' time -- of -- arrival observable. The problem of a time -- of -- arrival observable for the `waiting screen' is also considered. A method of avoiding the quantum Zeno effect is proposed and the positive operator valued measure (POV -- measure) or the generalized positive operator valued measure (GPOV -- measure) describing quantum time -- of -- arrival observable for the waiting screen are found.