Quantum time of arrival for the harmonic oscillator using the simplest symmetrization rule

Abstract

The quantum time of arrival for the harmonic oscillator Hamiltonian was obtained using the canonical commutation relation [H,T]=iℏI, where H is the Hamiltonian and T is the time of arrival operator. This was done using the basis operators S_m,n  from the simplest symmetrization rule, one of the different ordering rules along with Weyl and Born-Jordan ordering. The obtained time of arrival was found to be equivalent to the operator resulting from Weyl quantization. The resulting time kernels are also equal, regardless of whether simplest symmetric or Weyl quantization is used.