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CRC Press eBooks

Uncertainty

Jonathan Allday

Published Mar 3, 2009• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781584887041-20

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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.
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 Besides illustrating that the notion of a path is no longer valid in quantum mechanics, this cartoon also illustrates another significant difference between the quantum and classical worlds: the exact position of the white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.
According to the standard interpretation of quantum mechanics (QM), no meaning can be assigned to the statement that a particle has a precise value of any one of the variables describing its physical propertes before having interacted with a suitable measuring instrument. On the other hand, it is well known that QM tends to classical statistical mechanics (CSM) when a suitable classical limit is performed. One may ask therefore how is it that in this limit, the statement, meaningless in QM, that a given variable has always a precise value independently of having been measured, gradually becomes meaningful. In other words, one may ask how can it be that QM, which is a theory describing the intrinsically probabilistic properties of a quantum object, becomes a statistical theory describing a probabilistic knowledge of intrinsically well determined properties of classical objects.
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.
Quantum mechanical uncertainty After reviewing the main characteristics of the classical description of light, let us discuss those aspects of the quantization of the electromagnetic field which are of relevance for the analysis of the phenomena we are interested in. It seems a reasonable place to start to make clear the fundamental difference between the classical and the quantum mechanical description of nature; we will come across this difference many times when discussing experiments, and it will often give us a headache. We have to deal with the physical meaning of what is called uncertainty. The starting point of the classical description is the conviction that natural processes have a “factual” character. This means that physical variables such as the position or momentum of a particle have, in each single case, a well defined (in general, time dependent) value. However, it will not always be possible to measure all the appropriate variables (for instance, the instantaneous electric field strength of a radiation field); furthermore under normal circumstances we are able to measure only with a finite precision. Hence the basic credo of classical physics should be given in the following form: we are justified in imagining a world with variables possessing well defined values which are not known precisely (or not known at all). In doing this we are not forming any conclusions that contradict our everyday experiences. This is the fundamental concept of classical statistics: we are satisfied with the use of probability distributions for the variables we are interested in, not from fundamental but purely practical reasons.