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Corpus ID: 246015659

Time Operators of Harmonic Oscillators and Their Representations I

@inproceedings{Hiroshima2022TimeOO,
title={Time Operators of Harmonic Oscillators and Their Representations I},
author={Fumio Hiroshima and Noriaki Teranishi},
year={2022}
}

A time operator ˆ T ε of the one-dimensional harmonic oscillator ˆ h ε = 1 2( p 2 + εq 2 ) is rigorously constructed. It is formally expressed as ˆ T ε = 1 2 1 √ ε (arctan √ ε ˆ t + arctan √ ε ˆ t ∗ ) with ˆ t = p − 1 q . It is shown that the canonical commutation relation [ h ε , ˆ T ε ] = − i 1l holds true on a dense domain in the sense of sesqui-linear form, and the limit of ˆ T ε as ε → 0 is shown. Finally a matrix representation of ˆ T ε and its analytic continuation are given.

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