A self-adjoint arrival time operator inspired by measurement models

• Construction of a self-adjoint arrival time operator inspired by measurements.
• Agreement with the strong measurement formula in the low momentum regime.
• Review of self-adjoint and non-self-adjoint arrival time operators.
• Discussion of the momentum operator on the half-line.
• Discussion of the intuitive reasons obstructing self-adjointness.

We introduce an arrival time operator which is self-adjoint and, unlike previously proposed arrival time operators, has a close link to simple measurement models. Its spectrum leads to an arrival time distribution which is a variant of the Kijowski distribution (a re-ordering of the current) in the large momentum regime but is proportional to the kinetic energy density in the small momentum regime, in agreement with measurement models. A brief derivation of the latter distribution is given. We make some simple observations about the physical reasons for self-adjointness, or its absence, in both arrival time operators and the momentum operator on the half-line and we also compare our operator with the dwell time operator.