The time of arrival in quantum mechanics II. The individual measurement

Abstract

It is first shown that the time at which a moving classical particle reaches a fixed space point can be measured with arbitrarily high precision by the use of a suitably conceived apparatus. The same apparatus is then imagined to be exposed to a moving nonrelativistic quantum particle. Von Neumann's theory of quantum measurements is adapted to cover cases like this, in which pieces of information from the measurement process accumulate in the registration stages of the apparatus over an extended period of time. The relevant Schrödinger equation is solved. It is found that the apparatus fails to fulfill its designed purpose if its resolution time is too small. This implies an extension of the uncertainty principle and of the principle of complementarity—there are dynamical variables in classical particle mechanics which even singly admit no perfectly defined quantal counterparts. The status of the particle concept in the interpretation of nonrelativistic quantum mechanics is discussed in the light of these findings.