Time Operators of Harmonic Oscillators and Their Representations

A time operator T̂_ of the one-dimensional harmonic oscillator ĥ_=(p^2+ q^2) is rigorously constructed. It is formally expressed as T̂_=1/√() (arctan (√()t̂_0)+arctan (√()t̂_1)) with t̂_0=p^-1q and t̂_1=qp^-1. It is shown that the canonical commutation relation [h_, T̂_ ]=-i holds true on a dense domain in the sense of sesqui-linear forms, and the limit of T̂_ as → 0 is shown. Finally a matrix representation of T̂_ and its analytic continuation are given.