The von Neumann Model of Measurement in Quantum Mechanics

Jeffrey Bub

Pier A. Mello, Lars M. Johansen, Moises Martinez-Mares +1 more

Many textbooks on Quantum Mechanics are not very precise as to the meaning of making a measurement: as a consequence, they frequently make assertions which are not based on a dynamical description of the measurement process. A model proposed by von Neumann allows a dynamical description of measurement in Quantum Mechanics, including the measuring instrument in the formalism.In this article we apply von Neumann’s model to illustrate the measurement of an observable by means of a measuring instrument and show how various results, which are sometimens postulated without a dynamical basis, actually emerge.We also investigate the more complex, intriguing and fundamental problem of two successive measurements in Quantum Mechanics, extending von Neumann’s model to two measuring instruments. We present a description which allows obtaining, in a unified way, various results that have been given in the literature.

P. A. Moldauer

William Band

This paper attempts to explain both the basic concepts and the mathematical formalism of von Neumann's operator method in quantum statistics, in terms that should be understandable by anyone who has mastered first courses in quantum mechanics and statistical mechanics. A simple model is used throughout the paper to illustrate the various theorems and techniques. The discussion begins with the statistical interpretation of the Heisenberg uncertainty principle, the definition of mixed ensembles, and the meaning of measurement in quantum mechanics. The von Neumann projection operator is explained, and is used to set up the ensemble density matrix and the von Neumann entropy operator. In the second part of the paper, to be published later, the quantum-mechanical equation of change with time is applied to the von Neumann operators.

Rathindra Nath Sen

In his book Mathematical Foundations of Quantum Mechanics von Neumann asserted that the state vector must collapse upon measurement of any self-adjoint operator. This assertion was based on his reading of the Compton-Simon experiment. However, comparing von Neumann’s account with the Compton-Simon paper itself, we find that von Neumann had completely misinterpreted the experiment. Compton and Simon measured two angles on the same photographic plate; in von Neumann’s reading it became two successive measurements which gave identical results, where the experimenter could choose which measurement to perform first. In short, von Neumann’s case for collapse simply did not exist! In quantum mechanics, superpositions of different eigenvalues of conserved observables are freely admissible, but conservation laws require successive measurements of a conserved quantity upon the same object to return the same value (if the object is isolated and moves under a Schrödinger equation). This can be guaranteed only if the first measurement has caused the state vector to collapse. Thus collapse is a consistency condition which the theory must satisfy. Consistency problems are solved by displaying examples, or models, which fulfil the conditions in question. We therefore have to construct a mathematical model of the measurement process. This requires taking a stand on whether the measuring instrument does (Bohr) or does not (von Neumann-Wigner) have a classical description. If it does not, the ‘quantum measurement problem’ is insoluble. If it does, the problem – for conserved quantities – does have a solution, given by Sewell in 2005. The model displays collapse while evolving under the Schrödinger equation with a time-dependent hamiltonian; it does not have to call upon the observer’s ‘conscious ego’. Its existence shows that additive conservation laws are consistent with the superposition principle and Schrodinger evolution. Measurement of a conserved observable returns an eigenvalue of the observable; measurement of a non-conserved observable returns a diagonal matrix element of the observable in the basis of energy eigenvectors. If ‘classical descriptions’ of the apparatus are essential for these consistency conditions to be satisfied, what does it imply about the relation between classical and quantum mechanics? If the two are on the same footing, is it not possible that ‘dark matter’ is purely classical?

David E. Weeks