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?
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?
In his book `Mathematical Foundations of Quantum Mechanics', von Neumann asserted the following: the Compton-Simon experiment showed that the state vector must collapse upon measurement of any self-adjoint operator. Comparing von Neumann's account with the Compton-Simon paper, we find that von Neumann had misinterpreted the experiment as consisting of two successive measurements (which gave identical results), whereas the experiment only measured two angles on the same photographic plate. Note, however, that the state vector must collapse upon measurement of an additively-conserved quantity; otherwise the conservation law could be violated. Next, it turns out that the mathematical problem of explaining collapse is not fully defined until one specifies the nature of the apparatus. If the apparatus does not have a `classical description', the problem is insoluble, even if the measurement is only approximate (Fine, Simony, Brown, Simony and Busch); but if it does, the problem is soluble within Schroedinger dynamics (with a time-dependent hamiltonian) for additively-conserved observables. The solution, a modification of Sewell's, shows that the state vector has collapsed, but it does not reveal the eigenvalue of the collapsed state. The collapse is irreversible, and results from the interplay of additive conservation laws with the quantum measurement postulate. Indeed, the quantum measurement problem - as expounded by Wigner - may be better understood as the problem of establishing that the two are compatible with each other; it has little relevance to actual measurements.
Few things in physics have caused so much hand-wringing as von Neumann's collapse hypothesis. Unable to derive it mathematically, von Neumann attributed it to interaction with the observer's brain! Few physicists agreed, but tweaks of von Neumann's measurement theory did not lead to collapse, and Shimony and Brown proved theorems establishing `the insolubility of the quantum measurement problem'. Many different `interpretations' of quantum mechanics were put forward, none gained a consensus, and some scholars suggested that the foundations of quantum mechanics were flawed to begin with. Yet, in the last ninety years, no-one looked into now von Neumann had arrived at his collapse hypothesis! Von Neumann based his argument on the experiment of Compton and Simon. But, by comparing readings from von Neumann's book and the Compton-Simon paper, we find that the experiment provides no evidence for the collapse hypothesis; von Neumann had misread it completely! We suggest that von Neumann had relied on his phenomenal memory rather than the printed Compton-Simon paper, and his memory had failed him for once. Our finding has considerable implications for physics, which -- briefly sketched here -- will be discussed elsewhere in detail. An Appendix raises some questions for historians of physics.
Modified versions of the Schrödinger equation have been proposed in order to incorporate the description of measurement processes into the mathematical structure of quantum theory. Typically, these proposals introduce new physical constants, and imply small violations of momentum and energy conservation. These problematic features can be eliminated by assuming that wave function collapse is induced by the individual interactions that establish correlations between systems. The generation of a sufficient number of small, random shifts of amplitude between interacting and noninteracting branches of the wave function can bring about collapse on a scale consistent with our macroscopic experience. Two-particle interaction potential energies can be used as the basis for a collapse term added to the Schrödinger equation. The range of the interactions sets the distance scale of the collapse effects; the ratio of potential energies to the total relativistic energies of the particles determines the magnitude of the amplitude shifts, and the rate at which the interactions proceed fixes the timing parameters. Consistency with conservation laws in individual experiments is maintained because the collapse operator automatically takes into account the small, residual entanglement between the measured system and systems with which it has previously exchanged conserved quantities during interactions. Conservation is exact for momentum and orbital angular momentum, and it holds for energy within the accuracy allowed by the limited forms of energy describable in nonrelativistic theory.
Abstract In the context of the century-long debate on quantum measurement problem, the current work proposes a model that describes the process of collapse of state by quantum interaction, which resolves the controversies of the framework of quantum mechanics and describes the entire domain of quantum-to-classical world including the weak measurement and partial collapse. ‘Measurement’, being the process of physically interacting with a system in order to extracting information from it, is theorized in the current model by synthesizing the quantum interaction between system and measuring apparatus with the information entropy of such process. The model assumes Schrödinger equation to be the only guiding equation for all physical systems including the measuring apparatus, and does not presuppose ‘superposition principle’, rather derives it theoretically from the formulation. The superposed state is shown to be independent of the choice of measurement operator (observable) or basis states (pointers) of the measuring apparatus. Most interestingly, the current model explains the non-observance of ‘superposition principle’ by classical systems as the classical limit of such quantum description of measurement. Along with solving the quantum measurement problem, the work also explains weak measurement and partial collapse, which can be further extended to investigate such several emerging critical phenomena.
Abstract In the context of the century-long debate on quantum measurement problem, the current work proposes a model that describes the process of collapse of state by quantum interaction, which resolves the controversies of the framework of quantum mechanics and describes the entire domain of quantum-to-classical world including the weak measurement and partial collapse. ‘Measurement’, being the process of physically interacting with a system in order to extracting information from it, is theorized in the current model by synthesizing the quantum interaction between system and measuring apparatus with the information entropy of such process. The model assumes Schrödinger equation to be the only guiding equation for all physical systems including the measuring apparatus, and does not presuppose ‘superposition principle’, rather derives it theoretically from the formulation. The superposed state is shown to be independent of the choice of measurement operator (observable) or basis states (pointers) of the measuring apparatus. Most interestingly, the current model explains the non-observance of ‘superposition principle’ by classical systems as the classical limit of such quantum description of measurement. Along with solving the quantum measurement problem, the work also explains weak measurement and partial collapse, which can be further extended to investigate such several emerging critical phenomena.
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.