A pair of Hermitian operators is canonical if they satisfy the canonical commutation relation. It has been believed that no such canonical pair exists in finite-dimensional Hilbert space. Here, we obtain canonical pairs by noting that the canonical commutation relation holds in a proper subspace of the Hilbert space. For a given Hilbert space, we… Read More »
One of the fundamental problems in quantum mechanics is finding the correct quantum image of a classical observable that would correspond to experimental measurements. We investigate for the appropriate quantization rule that would yield a Hamiltonian that obeys the quantum analogue of Hamilton’s equations of motion, which includes differentiation of operators with respect to another… Read More »
It has been shown in Phys. Rev. Lett., \textbf{108} 170402 (2012), that quantum tunneling is instantaneous using a time-of-arrival (TOA) operator constructed by Weyl quantization of the classical TOA. However, there are infinitely many possible quantum images of the classical TOA, leaving it unclear if one is uniquely preferred over the others. This raises the… Read More »
Abstract The construction of time of arrival (TOA) operators canonically conjugate to the system Hamiltonian entails finding the solution of a specific second-order partial differential equation called the time kernel equation (TKE). In this paper, we provide an exact analytic solution of the TKE for a special class of potentials satisfying a specific separability condition.… Read More »
The role of conjugacy in the dynamics of time of arrival operators The construction of time of arrival (TOA) operators canonically conjugate to the system Hamiltonian entails finding the solution of a specific second-order partial differential equation called the time kernel equation (TKE). An expanded iterative solution of the TKE has been obtained recently in… Read More »
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… Read More »
The tunneling time problem earlier studied in Phys. Rev. Lett. \textbf{108}, 170402 (2012) using a non-relativistic time-of-arrival (TOA) operator predicted that tunneling time is instantaneous implying that the wavepacket becomes superluminal below the barrier. The non-relativistic treatment raises the question whether the superluminal behavior is a mere non-relativistic phenomenon or an an inherent quantum effect… Read More »
A time of arrival (TOA) operator that is conjugate with the system Hamiltonian was constructed by Galapon without canonical quantization in [J. Math. Phys. \textbf{45}, 3180 (2004)]. The constructed operator was expressed as an infinite series but only the leading term was investigated which was shown to be equal to the Weyl-quantized TOA-operator for entire… Read More »
A relativistic version of the Aharonov-Bohm time of arrival operator for spin-0 particles was constructed by Razavi in [Il Nuovo Cimento B \textbf{63}, 271 (1969)]. We study the operator in detail by taking its rigged Hilbert space extension. It is shown that the rigged Hilbert space extension of the operator provides more insights into the… Read More »
A time operator is a Hermitian operator that is canonically conjugate to a given Hamiltonian. For a particle in 1-dimension, a Hamiltonian conjugate operator in position representation can be obtained by solving a hyperbolic second-order partial differential equation, known as the time kernel equation, with some boundary conditions. One possible solution is the time of… Read More »