Momentum time of arrival

Abstract

We introduce the notion of the momentum time of arrival τ_p for the harmonic oscillator. We derive the classical momentum time of arrival and quantize it, obtaining the kernel of the integral form of the momentum time of arrival operator in its coordinate representation. We examine the symmetry satisfied by the kernel, eigenfunctions and eigenvalues. Using quadrature we evaluate the eigenvalue problem of this integral operator, and allow the eigenfunctions to evolve in time. We show that the variance of the associated probablility densities are minimum in the neighborhood of the momentum time of arrival eigenvalues, verifying the physical interpretation of the operator τ̂_p as a time of arrival operator.