Probability distribution of arrival times in quantum mechanics

In a previous paper [V. Delgado and J. G. Muga, Phys. Rev. A 56, 3425 (1997)] we introduced a self-adjoint operator $\mathcal{T}^\mathrm{}\left(X\right)\mathrm{}$ 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 $J^{\mathrm{}(+)\mathrm{}}\mathrm{}\left(X\right)\mathrm{}$, 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 $p_0\ne 0$, the expectation value of $J^{\mathrm{}(+)\mathrm{}}\mathrm{}\left(X\right)\mathrm{}$ 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.