Time Operators of Harmonic Oscillators and Their Representations
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929306594181120 |
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| author | Hiroshima, Fumio Teranishi, Noriaki |
| author_facet | Hiroshima, Fumio Teranishi, Noriaki |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2201_06352 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Time Operators of Harmonic Oscillators and Their Representations Hiroshima, Fumio Teranishi, Noriaki Mathematical Physics 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. |
| title | Time Operators of Harmonic Oscillators and Their Representations |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2201.06352 |