A Relativistic One-Particle Time of Arrival Operator for a Free Spin-1/2 Particle in (1 + 1) Dimensions

Abstract

As a follow-up to a recent study in the spin-0 case [J. Bunao and E. A. Galapon, Ann. Phys. 353, 83-106 (2015)], we construct a one-particle Time of Arrival (TOA) operator conjugate to a Hamiltonian describing a free relativistic spin-1/2 particle in one spatial dimension. Upon transformation in a representation where the Hamiltonian is diagonal, it turns out that the constructed operator consists of an operator term $\mathcal{\hat{T}}$ whose action is the same as in the spin-0 case, and another operator term $\mathcal{\hat{T}}_0$ which commutes with the Hamiltonian but breaks invariance under parity inversion. If we must impose this symmetry on our TOA operator, then we can throw away $\mathcal{\hat{T}}_0$ so that the TOA operator is just $\mathcal{\hat{T}}$.