The role of conjugacy in the dynamics of time of arrival operators

TL;DR

This paper derives an exact solution to the time kernel equation for certain potentials, enabling analysis of the dynamics of time of arrival operators and showing their unitary evolution and improved sharpness over quantized versions.

Contribution

It provides an exact analytic solution to the time kernel equation for a class of potentials, advancing understanding of TOA operator dynamics and conjugacy preservation.

Findings

CPTOA exhibits smoother, sharper unitary dynamics.

Eigenfunctions show unitary arrival at expected times.

Comparison favors CPTOA over Weyl-quantized operators.

Abstract

The construction of time of arrival (TOA) operators canonically conjugate to the system Hamiltonian entails finding the solution of a specific second-order partial differential equation called the time kernel equation (TKE). In this paper, we provide an exact analytic solution of the TKE for a special class of potentials satisfying a specific separability condition. The solution enables us to investigate the time evolution of the eigenfunctions of the conjugacy-preserving TOA operators (CPTOA) and show that they exhibit unitary arrival at the intended arrival point at a time equal to their corresponding eigenvalues. We also compare the dynamics between the TOA operators constructed by quantization and those independent of quantization for specific interaction potentials. We find that the CPTOA operator possesses smoother and sharper unitary dynamics over the Weyl-quantized one within numerical accuracy.