Characteristic time operators as quantum clocks
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912171072421888 |
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| author | Farrales, Ralph Adrian E. Galapon, Eric A. |
| author_facet | Farrales, Ralph Adrian E. Galapon, Eric A. |
| contents | We consider the characteristic time operator $\mathsf{T}$ introduced in [E. A. Galapon, Proc. R. Soc. Lond. A, 458:2671 (2002)] which is bounded and self-adjoint. For a semibounded discrete Hamiltonian $\mathsf{H}$ with some growth condition, $\mathsf{T}$ satisfies the canonical relation $[\mathsf{T},\mathsf{H}]|ψ\rangle=i\hbar|ψ\rangle$ for $|ψ\rangle$ in a dense subspace of the Hilbert space. While $\mathsf{T}$ is not covariant, we show that it still satisfies the canonical relation in a set of times of total measure zero called the time invariant set $\mathscr{T}$. In the neighborhood of each time $t$ in $\mathscr{T}$, $\mathsf{T}$ is still canonically conjugate to $\mathsf{H}$ and its expectation value gives the parametric time. Its two-dimensional projection saturates the time-energy uncertainty relation in the neighborhood of $\mathscr{T}$, and is proportional to the Pauli matrix $σ_y$. Thus, one can construct a quantum clock that tells the time in the neighborhood of $\mathscr{T}$ by measuring a compatible observable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_03364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characteristic time operators as quantum clocks Farrales, Ralph Adrian E. Galapon, Eric A. Quantum Physics We consider the characteristic time operator $\mathsf{T}$ introduced in [E. A. Galapon, Proc. R. Soc. Lond. A, 458:2671 (2002)] which is bounded and self-adjoint. For a semibounded discrete Hamiltonian $\mathsf{H}$ with some growth condition, $\mathsf{T}$ satisfies the canonical relation $[\mathsf{T},\mathsf{H}]|ψ\rangle=i\hbar|ψ\rangle$ for $|ψ\rangle$ in a dense subspace of the Hilbert space. While $\mathsf{T}$ is not covariant, we show that it still satisfies the canonical relation in a set of times of total measure zero called the time invariant set $\mathscr{T}$. In the neighborhood of each time $t$ in $\mathscr{T}$, $\mathsf{T}$ is still canonically conjugate to $\mathsf{H}$ and its expectation value gives the parametric time. Its two-dimensional projection saturates the time-energy uncertainty relation in the neighborhood of $\mathscr{T}$, and is proportional to the Pauli matrix $σ_y$. Thus, one can construct a quantum clock that tells the time in the neighborhood of $\mathscr{T}$ by measuring a compatible observable. |
| title | Characteristic time operators as quantum clocks |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2409.03364 |