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Communications in Mathematical Physics

The Time-Evolution of States in Quantum Mechanics according to the ETH-Approach

Jürg Martin Fröhlich, Alessandro Pizzo

Published Oct 23, 2021• 11 citations
Publisher

Abstract

Abstract It is argued that the Schrödinger equation does not yield a correct description of the quantum-mechanical time evolution of states of isolated physical systems featuring events. A general statistical law replacing unitary Schrödinger evolution of states is then formulated within the so-called ETH-Approach to Quantum Mechanics. This law eliminates the infamous “measurement problem.” Our general concepts and results are illustrated by an analysis of simple models describing a very heavy atom coupled to the quantized radiation field. In the limit where the speed of light tends to infinity these models can be treated quite explicitly.

DOI
10.1007/s00220-021-04225-5

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Abstract It is argued that the Schrödinger equation does not yield a correct description of the quantum-mechanical time evolution of states of isolated physical systems featuring events. A general statistical law replacing unitary Schrödinger evolution of states is then formulated within the so-called ETH-Approach to Quantum Mechanics. This law eliminates the infamous “measurement problem.” Our general concepts and results are illustrated by an analysis of simple models describing a very heavy atom coupled to the quantized radiation field. In the limit where the speed of light tends to infinity these models can be treated quite explicitly.
It is argued that the Schrödinger equation does not yield a correct description of the quantum-mechanical time evolution of states of isolated (open) systems featuring events. A precise general law for the time evolution of states replacing the Schrödinger equation is formulated within the so-called ETH-Approach to Quantum Mechanics. This law eliminates the infamous "measurement problem." Our general results are illustrated by an analysis of simple models describing a very heavy atom coupled to the quantized radiation field in a limit where the speed of light tends to infinity. The discussion of these models is the main subject of our paper.
This is a survey of a novel approach, called “ETH approach”, to the quantum theory of events happening in isolated physical systems and to the effective time evolution of states of systems featuring events. In particular, we attempt to present a clear explanation of what is meant by an “event” in quantum mechanics and of the significance of this notion. We then outline a theory of direct (projective) and indirect observations or recordings of physical quantities and events. Some key ideas underlying our general theory are illustrated by studying a simple quantum-mechanical model of a mesoscopic system.
This note starts with a recapitulation of what people call the ``Measurement Problem'' of Quantum Mechanics (QM). The dissipative nature of the quantum-mechanical time-evolution of averages of states over large ensembles of identical isolated systems consisting of matter interacting with the radiation field is discussed and shown to correspond to a stochastic time-evolution of states of individual systems. The importance of dissipation for the successful completion of measurements is highlighted. To conclude, a solution of the ``Measurement Problem'' is sketched in an idealized model of a double-slit experiment.
In this treatise I introduce the time dependent Generalized Born's Rule for the probabilities of quantum events, including conditional and consecutive probabilities, as the unique fundamental time evolution equation of quantum theory. Then these probabilities, computed from states and events, are to be compared with relative frequencies of observations. Schrodinger's equation still is valid in one model of the axioms of quantum theory, which I call the Schrodinger model. However, the role of Schrodinger's equation is auxiliary, since it serves to help compute the continuous temporal evolution of the probabilities given by the Generalized Born's Rule. In other models, such as the Heisenberg model, the auxiliary equations are quite different, but the Generalized Born's Rule is the same formula (covariance) and gives the same results (invariance). Also some aspects of the Schrodinger model are not found in the isomorphic Heisenberg model, and they therefore do not have any physical significance. One example of this is the infamous collapse of the quantum state. Other quantum phenomena, such as entanglement, are easy to analyze in terms of the Generalized Born's Rule without any reference to the unnecessary concept of collapse. Finally, this leads to the possibility of quantum theory with other sorts of auxiliary equations instead of Schrodinger's equation, and examples of this are given. Throughout this treatise the leit motif is the central importance of quantum probability and most especially of the simplifying role of the time dependent Generalized Born's Rule in quantum theory.
The postulates of von Neumann and Lüders concerning measurements in quantum mechanics are discussed and criticized in the context of a simple model proposed by Gisin. The main purpose of our paper is to analyze some mathematical aspects of that model and to draw some general lessons on the so-called ``measurement problem'' in quantum mechanics pointing towards the need to introduce general principles that determine the law for the stochastic time evolution of states of individual physical systems.
In order to describe the time evolution of energy states we choose to abandon the Hamiltonian method of quantum mechanics, which has been the standard for over a century, in favor of the more fundamental Lagrangian method. Integral equations of motion for the absorption and emission of radiation are derived to replace the differentially motivated Schrödinger equation. This new interpretation for the time evolution of energy states is used to interpret several well-known experiments, among them Planck’s law, the electron cyclotron, and chaos theory; thereby demonstrating an improved physical model of both microscopic and macroscopic phenomena.