Time-of-arrival distribution for arbitrary potentials and Wigner’s time-energy uncertainty relation

A realization of the concept of “crossing state” invoked, but not implemented, by Wigner, allows advancement in two important aspects of the formalization of the time of arrival in quantum mechanics: (i) For free motion, we find that the limitations described by Aharonov et al. in Phys. Rev. A 57, 4130 (1998) for the time-of-arrival uncertainty at low energies for certain measurement models are in fact already present in the intrinsic time-of-arrival distribution of Kijowski ${\Pi }_K;$ (ii) we have also found a covariant generalization of this distribution for arbitrary potentials and positions.