Quantum corrections to the Weyl quantization of the classical time of arrival

TL;DR

This paper analyzes quantum corrections to the Weyl-quantized time of arrival operator, explicitly solving all terms in its expansion and exploring their properties, especially in nonlinear systems like anharmonic oscillators.

Contribution

It provides a complete explicit solution for the TOA operator, including quantum corrections beyond the leading Weyl-quantized term, for general potentials.

Findings

Quantum corrections vanish for linear systems.

Quantum corrections are nonzero for nonlinear systems.

Explicit example for anharmonic oscillator analyzed.

Abstract

A time of arrival (TOA) operator that is conjugate with the system Hamiltonian was constructed by Galapon without canonical quantization in [J. Math. Phys. \textbf{45}, 3180 (2004)]. The constructed operator was expressed as an infinite series but only the leading term was investigated which was shown to be equal to the Weyl-quantized TOA-operator for entire analytic potentials. In this paper, we give a full account of the said TOA-operator by explicitly solving all the terms in the expansion. We interpret the terms beyond the leading term as the quantum corrections to the Weyl quantization of the classical arrival time. These quantum corrections are expressed as some integrals of the interaction potential and their properties are investigated in detail. In particular, the quantum corrections always vanish for linear systems but nonvanishing for nonlinear systems. Finally, we consider the case of an anharmonic oscillator potential as an example.