Conjugate Operators of 1D-harmonic Oscillator

TL;DR

This paper classifies conjugate operators of the 1D harmonic oscillator, explores their properties, and investigates the periodicity of their time evolution, providing a detailed mathematical framework for understanding these operators.

Contribution

It introduces a classification scheme for conjugate operators of the harmonic oscillator based on complex parameters and analyzes their time evolution behavior.

Findings

Classification of conjugate operators into three sets.

Explicit form of conjugate operators using logarithmic functions.

Demonstration of periodicity in the time evolution of these operators.

Abstract

A conjugate operator $T$ of one-dimensional harmonic oscillator $N$ is defined by an operator satisfying canonical commutation relation $[N,T]=-i\one$ on some domain but not necessarily a dense one. Examples of conjugate operators include the angle operator $\TA$ and the Galapon operator $\TG$. Let $\sT$ denote a set of conjugate operators of $N$ of the form $T_{\omega,m}=\frac{i}{m}\log(\omega\one-L^m)$ with $(\omega, m)\in \overline{\DD}\times (\NN\setminus\{0\})$, where $L$ is a shift operator and $\DD$ denotes the open unit disc in the complex plane $\CC$. A classification of $\sT$ is given as $\sT=\sT_{\{0\}}\cup\sT_{\DD\setminus\{0\}}\cup \sT_{\partial \DD}$, where $\TA\in\sT_{\{0\}}$ and $\TG\in \sT_{\partial \DD}$. The classification is specified by a pair of parameters $(\om,m)\in\CC\times\NN$. Finally the time evolution $T_{\om,m}(t)=e^{itN} T_{\om,m}e^{-itN}$ for $T_{\om,m}\in\sT$ is investigated, and it is shown that $T_{\om,m}(t)$ is periodic with respect to~$t$.