Free motion time-of-arrival operator and probability distribution.pdf

a r X i v : q u a n t - p h / 9 9 0 5 0 2 3 v 1 7 M a y 1 9 9 9 EHU-FT/9901, quant-ph/9905023 Free motion time-of-arrival operator and probability distribution I.L. Egusquiza1 and J.G. Muga2 1Fisika Teorikoaren Saila, Euskal Herriko Unibertsitatea, 644 P.K., 48080 Bilbao, Spain 2Departamento de F??sica Fundamental II, Universidad de La Laguna, La Laguna, Tenerife, Spain (February 1, 2008) We reappraise and clarify the contradictory statements found in the literature concerning the time- of-arrival operator introduced by Aharonov and Bohm in Phys. Rev. 122, 1649 (1961). We use Naimark’s dilation theorem to reproduce the generalized decomposition of unity (or POVM) from any self-adjoint extension of the operator, emphasizing a natural one, which arises from the analogy with the momentum operator on the half-line. General time operators are set within a unifying perspective. It is shown that they are not in general related to the time of arrival, even though they may have the same form. PACS: 03.65.-w I. INTRODUCTION Even though there is no question that experimental- ists measure distributions of time of arrival for quantum systems, there has been a long debate about the capa- bilities of standard quantum mechanics to address the very concept of time as an observable. In a nutshell, the problem arises because we expect observables to be represented in the quantum formalism by self-adjoint op- erators, whereas an old theorem of Pauli [1] states that for a semibounded self-adjoint operator H? no conjugate self-adjoint operator T? can exist, i.e., no operator that is self-adjoint and satisfies the canonical commutation re- lation [H?, T? ] = ih? over a dense domain. In other words, there is no self-adjoint time operator if the Hamiltonian is bounded from below, as we normally expect it to be. However the theorem has not discouraged theorists from attempting to fit such an immediate classical concept into the standard framework (see [2] for a general discussion). In