Solutions to the time-energy canonical commutation relation using Weyl, Symmetric, and Born-Jordan basis operators

Abstract

Quantization of the classical time of arrival often fails to preserve the required commutator algebra of the operators, given by [H,T]=iħ1, where H is the Hamiltonian and T is the time of arrival operator. In the free particle system, quantization of the local time of arrival leads to the correct time operator. In the quantum free-fall and harmonic oscillator system, Weyl quantization of the classical time of arrival results to time operators that follow the necessary commutation relation, while symmetric and Born-Jordan quantization fails to do so. This problem is addressed by finding the minimal solution of the time-energy canonical commutation relation in Weyl, symmetric, and Born-Jordan basis. We use this method in solving the time operator for the harmonic oscillator.