Moyal deformation of the classical arrival time

• URL: http://arxiv.org/abs/2309.00222v4
• Date: Fri, 27 Sep 2024 00:48:27 GMT
• Title: Moyal deformation of the classical arrival time
• Authors: Dean Alvin L. Pablico, Eric A. Galapon,
• Abstract summary: We find an appropriate quantum image of the classical arrival time $mathcalT_C(q,p)$, usually in operator form $hatmathrmT$. The resulting quantum image is a real-valued and time-reversal symmetric function $mathcalT_M(q,p)$ in formal series of $hbar2$ with the classical arrival time as the leading term.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The quantum time of arrival (TOA) problem requires the statistics of measured arrival times given only the initial state of a particle. Following the standard framework of quantum theory, the problem translates into finding an appropriate quantum image of the classical arrival time $\mathcal{T}_C(q,p)$, usually in operator form $\hat{\mathrm{T}}$. In this paper, we consider the problem anew within the phase space formulation of quantum mechanics. The resulting quantum image is a real-valued and time-reversal symmetric function $\mathcal{T}_M(q,p)$ in formal series of $\hbar^2$ with the classical arrival time as the leading term. It is obtained directly from the Moyal bracket relation with the system Hamiltonian and is hence interpreted as a Moyal deformation of the classical TOA. We investigate its properties and discuss how it bypasses the known obstructions to quantization by showing the isomorphism between $\mathcal{T}_M(q,p)$ and the rigged Hilbert space TOA operator constructed in [Eur. Phys. J. Plus \textbf{138}, 153 (2023)] which always satisfy the time-energy canonical commutation relation (TECCR) for arbitrary analytic potentials. We then examine TOA problems for a free particle and a quartic oscillator potential as examples.

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