Time Operators of Harmonic Oscillators and Their Representations

TL;DR

This paper rigorously constructs a time operator for the harmonic oscillator, demonstrating its properties, commutation relations, and matrix representations, including the limit as the oscillator parameter approaches zero.

Contribution

It introduces a rigorous construction of the time operator for harmonic oscillators and explores its properties and representations, including the limit case.

Findings

The time operator satisfies the canonical commutation relation on a dense domain.

A matrix representation and analytic continuation of the time operator are provided.

The limit of the time operator as the oscillator parameter approaches zero is established.

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 $\hat t_1=qp^{-1}$. 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.