TIME OF ARRIVAL IN QUANTUM MECHANICS .I. FORMAL CONSIDERATIONS

It is argued that the complementarity of time and energy in one-dimensional wave mechanics finds its proper application in the measurement of the time at which a moving particle arrives at (or departs from) a fixed spatial point. A rigorous proof is given that the ordinary Hilbert space of a freely moving particle does not contain a set of measurement eigenstates for this measurement problem. The proof depends upon the semi-infinite extent of the energy spectrum. Both this result and the physical nature of an arrival-time situation suggest that the most suitable and general wave function for the problem might be obtained by arranging that the particle be emitted by a source situated at a finite distance from the point of detection. Such sources are introduced, and the motion over the intervening space is treated by using the one-dimensional nonrelativistic Schrödinger equation. The resultant free-particle wave functions contain components with both positive and negative energies, and constitute a linear space different from the ordinary Hilbert space. The arrival-time problem then becomes one of interpreting these unfamiliar wave functions, with a view to obtaining a formula for the arrival probability per unit time. It is proved with complete generality and rigour that this modified problem also has no solution. The mathematical properties of the time-dependent Schrödinger equation make it impossible to construct any operationally meaningful and apparatus-independent probability formula. This demonstrates the existence of genuine measurement problems which lie beyond the scope of the usual measurement formalism. Such problems can only be satisfactorily discussed by making reference to the mechanical parameters of the measuring apparatus. © 1969.