Quantizations of the classical time of arrival and their dynamics

TL;DR

This paper develops a method to quantize the classical time of arrival in quantum mechanics, exploring different ordering rules and analyzing the dynamics of the resulting operators' eigenfunctions.

Contribution

It introduces a generalized quantization framework for the time of arrival operator, including new constructions beyond standard ordering rules.

Findings

Eigenfunctions show unitary arrival at the target point.

Different quantization schemes affect the operator properties.

The approach allows constructing new time of arrival operators.

Abstract

The classical time of arrival in the interacting case is quantized by way of quantizing its expansion about the free time of arrival. The quantization is formulated in coordinate representation which represents ordering rules in terms of two variable polynomial functions. This leads to representations of the quantized time of arrival operators as integral operators whose kernels are determined by the chosen ordering rule. The formulation lends itself to generalization which allows construction of time of arrival operators that cannot be obtained by direct quantization using particular ordering rules. Wey, symmetric and Born-Jordan quantizations are specifically studied. The dynamics of the eigenfunctions of the different time of arrival operators are investigated. The eigenfunctions exhibit unitary arrival at the intended arrival point at their respective eigenvalues.