A one-particle time of arrival operator for a free relativistic spin-0 charged particle in (1+1) dimensions

Abstract

We construct a one-particle TOA operator $\hat{\mathcal{T}}$ canonically conjugate with the Hamiltonian describing a free, charged, spin-$0$, relativistic particle in one spatial dimension and show that it is maximally symmetric. We solve for its eigenfunctions and show that they form a complete and non-orthogonal set. Plotting the time evolution of their corresponding probability densities, it implies that the eigenfunctions become more localized at the origin at the time equal to their eigenvalues. That is, a particle being described by an eigenfunction of $\hat{\mathcal{T}}$ is in a state of definite arrival time at the origin and at the corresponding eigenvalue. We also calculate the TOA probability distribution of a particle in some initial state.

Preprint. Please refer to the Annals of Physics for the published version

References in corpus (2)

• Nonperturbative Quantum Gravity
• The Bender-Dunne basis operators as Hilbert space operators

Cited by in corpus (4)

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• Synchronizing quantum and classical clocks made of quantum particles
• Spacetime Quanta? : The Discrete Spectrum of a Quantum Spacetime Four-Volume Operator in Unimodular Loop Quantum Cosmology