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Journal of Mathematical Physics

Mathematics of the Measurement Problem in Quantum Mechanics

J. H. Tutsch

Published Aug 1, 1971• 6 citations
Publisher

Abstract

The mathematical structure of the simplest nontrivial hidden variable theory is derived. The core of the resulting theory is an essentially unique nonlinear differential equation which gives a causal description of the continuous collapse of the wavefunction during a measurement. It is shown that the simplest collapse equation is the same as the equation given ad hoc by Bohm and Bub.

DOI
10.1063/1.1665795

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The mathematical structure of the simplest nontrivial hidden variable theory is derived. The core of the resulting theory is an essentially unique nonlinear differential equation which gives a causal description of the continuous collapse of the wavefunction during a measurement. It is shown that the simplest collapse equation is the same as the equation given ad hoc by Bohm and Bub.
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A recently proposed theory of measurement for nonrelativistic quantum mechanics, due to Bohm and Bub, is generalized, and a mathematical analysis of the theory is presented. The fact that there are always theories of the Bohm-Bub type, given the standard formulation of quantum mechanics, is discussed. The phenomenological coefficient $\ensuremath{\gamma}$ of the original theory is defined to be the absolute value of the energy change of the system during measurement divided by $\ensuremath{\hbar}$. This definition leads to a Heisenberg-like relation between the average energy change of the system during the measurement process and the collapse time, but the new relation depends on the so-called hidden variables, and hence could lead to non-quantum-mechanical effects. The time dependence of the hidden variables is considered, and they are then assumed to be constant for each individual system. An experiment is proposed which would test this assumption. A positive result would disagree significantly with the quantum-mechanical prediction. The question of whether or not the new variables must remain unobservable is analyzed mathematically. A rigorous definition of simultaneous measurement is made, and the resulting interference is given a mathematical description.
Quantum physics confronts contemporary metaphysicians with a fascinating and frustrating puzzle. It appears to be telling us something novel and surprising about the nature of the physical world, but it is hard to pin down exactly what it tells us. Hidden variable theories constitute a second major class of solutions to the measurement problem. They differ from the standard theory and spontaneous collapse theories in eschewing collapse altogether; according to hidden variable theories, the wavefunction always evolves according to the Schrodinger equation. The simplest and most radical response to the measurement problem is to assert that standard quantum mechanics without wavefunction collapse constitutes a complete physical theory. The many worlds theory is more flexible in this regard; according to the standard contemporary version, no presupposition is made concerning precisely which physical properties form the basis of measurement results. There are several approaches to quantum mechanics, any or none of which might yield an acceptable physical theory.
Abstract The measurement problem in quantum mechanics still appears to be an unresolved issue. Here we present a new quantum theory of measurement that overcomes many of the difficulties previously found. It is based on a consistent use of the linear superposition principle and distinguishes two aspects: recording and observation. A recording elicits the full interaction of the object quantum system with the quantum measuring apparatus. No wavefunction collapse is introduced. Statistics may appear at the observation of the recording only and depends on filtering processes. The theory presented here uses the existing mathematical structure of quantum mechanics but requires no ad hoc measurement postulates. Well‐known paradoxical aspects in standard quantum mechanics, for instance, wave–particle duality, Schrödinger's cat, and Zeno effects do not appear in the current formulation. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
The main distinction between classical mechanics and quantum mechanics is the lack in the latter of a full mechanical determinism: different final states can arise from the same physical state, after the measurement. No hidden variable is supposed to exist, nothing can discriminate two apparently identical states even if they give a different result. In this paper we try to put the basis for a more fundamental theory that (approximately) coincides with quantum mechanics when comparing statistics, but it is more fundamental, since it mathematically describes measurement processes giving an explicit time evolution of the wave function during the collapse. The theory is deterministic even if the Heisenberg uncertainty principle is still valid. The theory distinguishes physical states that collapse and physical states that do not collapse. The theory can be made compatible with all experiments done in the past, but new phenomena such as violations of the Born law or the superposition principle could transpire. However, even if we have probably shown that it is possible to build ad hoc a theory that can describe both the wave function collapse and the Schrödinger linear evolution, a simple and unified construction is still missing.