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 𝒯(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 𝒯(q,p) appears as an infinite series expanded in even powers of ħ with the classical arrival time 𝒯₀(q,p) appearing only as the leading term. We then show that the Weyl quantization of 𝒯(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.