Time Operators of Harmonic Oscillators and Their Representations

Abstract

A time operator $\hat T_\eps$ of the one-dimensional harmonic oscillator $ \hat h_\eps=\half(p^2+\eps q^2)$ is rigorously constructed. It is formally expressed as $ \hat T_\eps=\half\frac{1}{\sqrt \eps } (\arctan (\sqrt \eps \hat t_0)+\arctan (\sqrt \eps \hat t_1))$ with ${\hat{t}}_0=p^{-1}q$ and ${\hat{t}}_1=q{p}^{-1}$. It is shown that the canonical commutation relation $[h_\eps, \hat T_\eps ]=-i\one$ holds true on a dense domain in the sense of sesqui-linear forms, and the limit of $\hat T_\eps $ as $\eps\to 0$ is shown. Finally a matrix representation of $\hat T_\eps$ and its analytic continuation are given.