Moyal deformation of the classical arrival time
- URL: http://arxiv.org/abs/2309.00222v2
- Date: Mon, 9 Oct 2023 04:49:58 GMT
- Title: Moyal deformation of the classical arrival time
- Authors: Dean Alvin L. Pablico and 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.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: The quantum time of arrival (TOA) problem requires statistics of the 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.
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