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CRC Press eBooks

Uncertainty

Jonathan Allday

Published Sep 21, 2022• 0 citations
Publisher

Abstract

In classical physics, the width, or uncertainty, in a set of experimental results is generally attributable to experimental variations, due to small differences in the objects being tested, or variations in the apparatus used, instrumental factors, environmental factors etc. Experiments in quantum physics also suffer from all these sorts of variations. Consequently, even if we have a collection of systems, all carefully prepared in the same eigenstate of the physical variable in question, we will not always get exactly the same measured value. The spread of this distribution is down to experimental uncertainty. People have not been afraid of drawing philosophical conclusions from the uncertainty principle, almost from the moment the result was first published. Heisenberg himself launched one of the earliest sallies, his target being the principle of causality. Causality, the notion that everything that happens has been caused by something happening previously, has been central to physics for hundreds of years.

DOI
10.1201/9781003225997-15

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In classical physics, the width, or uncertainty, in a set of experimental results is generally attributable to experimental variations, due to small differences in the objects being tested, or variations in the apparatus used, instrumental factors, environmental factors etc. Experiments in quantum physics also suffer from all these sorts of variations. Consequently, even if we have a collection of systems, all carefully prepared in the same eigenstate of the physical variable in question, we will not always get exactly the same measured value. The spread of this distribution is down to experimental uncertainty. People have not been afraid of drawing philosophical conclusions from the uncertainty principle, almost from the moment the result was first published. Heisenberg himself launched one of the earliest sallies, his target being the principle of causality. Causality, the notion that everything that happens has been caused by something happening previously, has been central to physics for hundreds of years.
Whenever physicists think about a perfect set of experimental values, they dream of something called the normal distribution curve, as illustrated in Figure 13.1. In this case we are supposed to imagine that Figure 13.1 represents the results of identical measurements made on a collection of objects. We are not just talking about quantum systems being measured here. e normal distribution applies to the experimental results found in classical physics as well, although for di erent physical reasons.
In a recent number of Blackfriars (July, 1949) Dr A. W. Gledhill has discussed very competently some aspects of the statistical method as used in modern physics. He has tried to show there ‘that modern physical theory does not contradict the principle of causality’. Having made a study of this subject over quite a number of years, the writer, while agreeing with some of Dr Gledhill’s conclusions, feels nevertheless that there are a few points which have not been covered as adequately as the importance of the subject for modern philosophy seems to warrant. In order to understand the controversy about the validity or non-validity of the principle of causality, which has been initiated by Heisenberg’s uncertainty principle, it is necessary to realise, first of all, that when a physicist speaks about causality, he usually does not mean the same thing as a scholastic philosopher. In scholastic philosophy one distinguishes different kinds of causes, such as the material, the efficient, the final, as well as the formal and exemplary causes. The classical physicist, on the other hand, since the time of Galileo, admits only efficient causes. Let us therefore give a brief sketch of the development of this concept of scientific causality and show the reasons for this limitation. The purpose of a scientific theory is the prediction of events among the natural objects which form the field of enquiry of the particular science concerned. Evidently, the more certain the prediction, the better the theory on which it is based.
The uncertainty principle, formulated by Heisenberg in 1927, is a fundamental principle in quantum mechanics. It forces us to review key concepts of classical physics, and from this analysis are born the concepts of quantum physics. Consequences of the Uncertainty principle also introduces the concept of incompatible sizes, a completely new concept compared to classical physics. A state of the system, which we call a quantum state, will be determined from the knowledge of a set maximum of certain quantities that are independent and compatible with each other. A measurable physical quantity is called an observable. If at a given moment a measurement is performed by one or more compatible observables it is said they are performing an observation of the system. The basic principles of quantum mechanics are expressed by a number of postulates. Their numbers and can vary from one text to another.
A philosopher once said ‘It is necessary for the very existence of science that the same conditions always produce the same results’. Well, they don't! Richard Feynman Watching electrons We have seen that quantum mechanics does not allow us the comfort of being able to visualize the motion of a quantum particle. In a normal game of billiards we can imagine the paths taken by the individual balls (Fig. 1.1). Figure 2.1 shows the physicist George Gamow's attempt to give some impression of how the same game might look if played with quantum particles. Besides illustrating that the notion of a path is no longer valid in quantum mechanics, this cartoon also illustrates another significant difference between the quantum and classical worlds: the exact position of the white ball is not known. Uncertainty has entered physics and replaced the determinism of Newtonian mechanics. By the nineteenth century, physicists had been able to explain vast amounts of experimental observations on objects as different as planets and billiard balls. If an observation differed from the predictions of classical physics, they looked for something they had overlooked to explain the deviation. In 1864, physicists' confidence in the whole edifice of classical physics seemed to be spectacularly verified by an analysis of some irregularities in the orbit of Uranus. These were attributed to the existence of a then undiscovered planet – the subsequent discovery of Neptune was a triumph for Newtonian physics.
On the level of quantum phenomena, there is no precise possibility of separating an examined physical system from a measuring instrument. This fact lends support to the assertion that the evolution of a quantum system ceases to be rigorously causal from the moment when the system is subjected to investigation. A quantum measurement is special in that it always causes a jump in the system under study to an eigenstate of the considered variable. Possible interactions between the measuring and measured objects will be neglected, so that a given quantum system, which is free from any disturbances, e.g. measurements or other perturbations, will evolve in a rigorously predictable manner. Globally speaking, a quantum-mechanical scheme of description of physical phenomena can be summarized by introducing the so-called Schrödinger picture. A physical field is obtained when a given physical feature is attributed to each point of a space in which the process under study is developing.
Identity principle is a fundamental postulate of quantum mechanics. In this paper, from a philosophical point of view, we revisit the identity principle of quantum mechanics. We attribute the identity to the existence of a precision limit for observation: the microscopic objects become indistinguishable when the difference between them is lower than the precision limit. This precision limit is determined not by the measuring apparatus, but by the action quantum h behand the objective physical law. We deduce logically how to relate the identity principle with the probabilistic interpretation, uncertainty principle, quantization in quantum mechanics, attempting to sorting out some logic for quantum theory. The indistinguishability of identical particles makes it impossible to accurately obtain all the details about the microscopic objects at the same time, and consequently the microscopic objects can only be described statistically, leading to the probabilistic determinism and indeterminacy of initial state in quantum mechanics. By reviewing Bohmian mechanics, one of the nonlocal hidden variable theories, we discuss the possibility of restoring determinism and causality in quantum mechanics by restoring the determinacy of initial state. However, any hidden variable theory constructed by using only the terminologies of classical physics cannot reproduce the quantum mechanical predictions. It is the identity principle that makes it impossible to restore causality and determinism, and is the root that makes quantum mechanics different from classical physics. Finally, the boundary between the so-called classical and quantum worlds is discussed. We hope these philosophical arguments can provide a kind of epistemic understanding for quantum mechanics.