The role of conjugacy in the dynamics of time of arrival operators
arxiv(2024)
摘要
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 [Eur.
Phys. J. Plus 138, 153 (2023)] but is generally intractable to be
useful for arbitrary nonlinear potentials. In this work, we provide an exact
analytic solution of the TKE for a special class of potentials satisfying a
specific separability condition. The solution enables us to investigate the
time evolution of the eigenfunctions of the conjugacy-preserving TOA operators
(CPTOA) by coarse graining methods and spatial confinement. We show that the
eigenfunctions of the constructed operator exhibit unitary arrival at the
intended arrival point at a time equal to their corresponding eigenvalue.
Moreover, we examine whether there is a discernible difference in the dynamics
between the TOA operators constructed by quantization and those independent of
quantization for specific interaction potentials. We find that the CPTOA
operator possesses better unitary dynamics over the Weyl-quantized one within
numerical accuracy. This allows us determine the role of the canonical
commutation relation between time and energy on the observed dynamics of time
of arrival operators.
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