Teaching ideal quantum measurement, from dynamics to interpretation

Abstract

We present a graduate course on ideal measurements, analyzed as dynamical processes of interaction between the tested system S and an apparatus A, described by quantum statistical mechanics. The apparatus A = M + B involves a macroscopic measuring device M and a bath B. The requirements for ideality of the measurement allow us to specify the Hamiltonian of the isolated compound system S + M + B . The resulting dynamical equations may be solved for simple models. Conservation laws are shown to entail two independent relaxation mechanisms: truncation and registration. Approximations, justified by the large size of M and of B, are needed. The final density matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>𝒟</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> ( t f ) of S + A has an equilibrium form. It describes globally the outcome of a large set of runs of the measurement. The measurement problem, i.e., extracting physical properties of individual runs from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>𝒟</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> ( t f ), then arises due to the ambiguity of its splitting into parts associated with subsets of runs. To deal with this ambiguity, we postulate that each run ends up with a distinct pointer value A i of the macroscopic M. This is compatible with the principles of quantum mechanics. Born’s rule then arises from the conservation law for the tested observable; it expresses the frequency of occurrence of the final indications A i of M in terms of the initial state of S. Von Neumann’s reduction amounts to updating of information due to selection of A i . We advocate the terms q -probabilities and q -correlations when analyzing measurements of non-commuting observables. These ideas may be adapted to different types of courses.

Topics & Concepts