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arXiv:quant-ph/9807043 · Physical Review A

Time-of-arrival states

Jonathan Oppenheim, Benni Reznik, William George Unruh

Published Mar 1, 1999• 22 citations
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Abstract

Although one can show formally that a time-of-arrival operator cannot exist, one can modify the low-momentum behavior of the operator slightly so that it is self-adjoint. We show that such a modification results in the difficulty that the eigenstates are drastically altered. In an eigenstate of the modified time-of-arrival operator, the particle, at the predicted time of arrival, is found far away from the point of arrival with probability $1/2.$

DOI
10.1103/physreva.59.1804

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Although one can show formally that a time-of-arrival operator cannot exist, one can modify the low-momentum behavior of the operator slightly so that it is self-adjoint. We show that such a modification results in the difficulty that the eigenstates are drastically altered. In an eigenstate of the modified time-of-arrival operator, the particle, at the predicted time of arrival, is found far away from the point of arrival with probability 1/2.1/2.1/2.
In a previous paper [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997)] we introduced a self-adjoint operator T(X)\mathcal{T}^(X)T(X) whose eigenstates can be used to define consistently a probability distribution of the time of arrival at a given spatial point. In the present work we show that the probability distribution previously proposed can be well understood on classical grounds in the sense that it is given by the expectation value of a certain positive-definite operator J(+)(X){J}^{(+)}(X)J(+)(X), which is nothing but a straightforward quantum version of the modulus of the classical current. For quantum states highly localized in momentum space about a certain momentum ${p}_{0}\ensuremath{\ne}0$, the expectation value of J(+)(X){J}^{(+)}(X)J(+)(X) becomes indistinguishable from the quantum probability current. This fact may provide a justification for the common practice of using the latter quantity as a probability distribution of arrival times.
We show that formulating the quantum time of arrival problem in a segment of the real line suggests rephrasing the quantum time of arrival problem to finding states that evolve to unitarily collapse at a given point at a definite time. For the spatially confined particle, we show that the problem admits a solution in the form of an eigenvalue problem of a compact and self-adjoint time of arrival operator derived by a quantization of the classical time of arrival, which is canonically conjugate with the Hamiltonian in a closed subspace of the Hilbert space.