A time of arrival operator for a quartic oscillator potential from the Moyal bracket formalism

Abstract

We construct a phase space time of arrival (TOA) function ð’¯(q,p) for the case of a quartic oscillator potential from its Moyal backet with the system Hamiltonian in quantum phase space, instead of the usual Poisson bracket in classical mechanics. We show that ð’¯(q,p) appears as an infinite series expanded in even powers of ħ with the classical arrival time ð’¯₀(q,p) appearing only as the leading term. We then show that the Weyl quantization of ð’¯(q,p) is exactly the supraquantized TOA operator for a quartic oscillator. This strongly suggests the possibility of using canonical quantization to construct TOA operators that generally satisfy the conjugacy requirement with the system Hamiltonian.