In this work I propose an analogy between Pythagoras's theorem and the logical-formal structure of Werner Heisenberg's “relations of uncertainty.” The reasons that they have pushed to me to place this analogy have been determined from the following ascertainment: Often, when in exact sciences a problem of measurement precision arises, it has been resolved with the resource of the elevation to the square. To me it seems also that the aporie deriving from the uncertainty principle can find one solution with the resource to this stratagem. In fact, if the first classic example of the argument is the solution of the incommensurability between catheti and the hypotenuse of the triangle rectangle, one of the last cases is that which is represented from Heisenberg's principle of uncertainty.
In this work I propose an analogy between Pythagoras's theorem and the logical-formal structure of Werner Heisenberg's “relations of uncertainty.” The reasons that they have pushed to me to place this analogy have been determined from the following ascertainment: Often, when in exact sciences a problem of measurement precision arises, it has been resolved with the resource of the elevation to the square. To me it seems also that the aporie deriving from the uncertainty principle can find one solution with the resource to this stratagem. In fact, if the first classic example of the argument is the solution of the incommensurability between catheti and the hypotenuse of the triangle rectangle, one of the last cases is that which is represented from Heisenberg's principle of uncertainty.
Abstract 1. In the general synthesis of classical dynamics with the quantum theory, the Uncertainty Principle plays a most useful part. It is of course only one aspect of the new mechanics, but it is a very helpful one since by its means it becomes easy to see where the old classical ideas broke down. The state of affairs in the quantum theory is not unlike that of the early days of relativity, when most of those who studied the subject felt the need of supporting the formal theory by seeing how the old ideas failed in specific cases. Here the formal theory is very abstract and is not easy to follow intuitively, and the Uncertainty Principle plays much the same rôle as did the examples of clocks and rods in relativity theory. For this reason it is more appropriate for illustrative examples, than for any extreme generality, and though a number of examples have been already given by Heisenberg and Bohr, it may not be amiss to have some more. There are probably some who will have shared my experience that it is often by no means easy to detect how the uncertainty enters into a given experiment, though once detected the arguments are usually very simple. In a recent conversation Professor Bohr criticised some rather careless remarks that I had made, and the present work was undertaken to clear matters up. It may be shortly described as a study of the Uncertainty Principle in connection with electrometers and magnetometers. It may be of interest to point out that the Uncertainty Principle can be regarded from a rather different aspect. The "resolving power" of optical instruments was discussed very fully by Rayleigh on wave principles, but as long as matter was regarded in the classical manner, a mechanical instrument could be considered as capable of measuring quantities with absolute accuracy. Now that we know that matter also has wave properties, there is need of a theory of the resolving power of mechanical instruments, and this is exactly what the Uncertainty Principle supplies.
Abstract One of the foundational principles of quantum mechanics is Heisenberg’s uncertainty relation. According to this relation, no one can measure both the position of a particle and how fast it is moving simultaneously no matter how precise the measuring instruments. Heisenberg’s uncertainty relation and the associated Bohr’s principle of complementarity are explained using simple examples. The idea that measurement disturbs the system is a source of the uncertainty. The philosophical question whether the uncertainty represents only our inability to measure the properties precisely or it has much deeper consequences related to the question of reality is also discussed.
Two years ago I indicated in this journal* [1] that the conclusions drawn from the formalism of quantum mechanics, and in particular from the uncertainty relations, [2] do not seem to necessarily indicate a failure of the causality principle, because the conceptual structures on which these conclusions are based stem from Newton’s mechanics and, like all of classical mechanics, are of empirical origin. In my opinion this reasoning only refutes applying those [classical] concepts to atomic processes. There is one argument in the discussion of this subject, however, that I had not commented on at the time, which initially sounds quite convincing. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The principles of relativity and uncertainty represent two of the deepest and most encompassing propositions of the physical sciences. Indeed, much of our present knowledge of nature can be recapitulated in these important statements about the processes of motion and measurement. However, a question remains as to the precise logical connection between both principles. Here, we show a plausible and simple conceptual analysis linking the principles of relativity and uncertainty as logical requirements of the idea of physical measurement. The main conclusion points to the logical complementarity of both principles in order to justify the possibility of physical measurements, in which the uncertainty principle guarantees the applicability of the principle of relativity in all physically conceivable systems of reference.
COGNITIVE ASPECTS OF THE HEISENBERG PRINCIPLE I T HE Heisenberg uncertainty principle, like its rival theory of relativity/ has been one of the truly crucial phases in the physics of our day~ It did not, like the discovery of a new element or the Maxwell equations, simply indent the story of classical physics for a new paragraph. It was not only a new chapter but a new book. Detected, in the words of Heisenberg himself, when physics became self-critical, the principle is a statement of what is observable-like its contemporary, the theory of relativity. It was anew advance in method, so revolutionary indeed that physics has virtually lost meaning as content and become synonymous , as logical empiricism amply indicates, with mere rules of procedure. That science is developing into a system of logic, combining the Cartesian stress on clearness with the Kantian regulative idea, actually confirms a contention of traditional philosophy. It reveals that once we deny the colors and sounds, the life, lights, and purpose in our universe-as pure scientific method actually does-we are fated to deny much more than quality. Quality is in the order of form, substance, and being. The moment quality is banished from the real world, by the same 1 That all is not as peaceful and unified in physics as the layman sometimes believes can be seen in the deep cleavage between quantum and relativity mechanics. For the first, the exact specification of space (position) and time (velocity, momentum) has no meaning, while in relativity theory such exact description is the only meaning that can be had. Despite the efforts 'of leading physicists, including Dirac, to unite the two theories, the contradiction is so profound that one of these systems must eventually be abandoned or, more probably, both of them will be superseded. The recent work .of Schroedinger and apparently also of Einstei~ himself are directed to extend the general theory of relativity, making the current theory a special case just as Newtonian physics is a special case of present relativity mechanics. 474 COGNITIVE ASPECTS OF THE HEISENBERG PRINCIPLE 475 decree being itself-because it is an immediate, non-inertial (formal) thing like quality-is driven into its only possible exile, nothingness. It is not surprising then that, after modern physics· has finished its task, the universe should not only be emptied of its lights, its fragrance, its sweetness, and its music but should also be without being and without existence. The uncertainty principle has been much debated in modern thought. Its impact has brought a sad end to the pride of 19th century science since it prevents a complete knowledge of anything at all by shading into the darkness at least one of the two major factors that specify it. Shortly after the principle was born it was hailed by various authors as a proof of free will, of the possibility of religion, and of non-mechanistic concepts in biology. In a more consistent and challenging sense, it has often been taken to refute what the physicist calls causality and what the philosopher calls the order of nature. The fact of the matter is that if the senses alone supply knowledge, truly only one thing at a time can be apprehended. If Hume was right in philosophy, so is Heisenberg. But when sensism yields to an intellectual insight into reality, it is easy to refute the scientist's claim to dethrone the doctrine of nature's order which he could never detect, even in classical physics, by pointerreadings .2 But whatever the fate of the Heisenberg principle in cosmology , it has had a most fruitful career in the field of physics. To Heisenberg himself, the principle was a point of departure for an original development of quantum mechanics based on the " uncertainty " of position and momentum,.: 3 If, for example, measurements are made, under a microscope, of the position of an electron, the photon (light-particle), which impinges on the eye and thus conveys the required knowledge to the physicist, has had to collide with the electron, thus imparting an impact known as the Compton recoil. When • For ·a discussion of the order of nature, cf. J. Marling: The...
Uncertainty relations are commonly praised as one of the central pillars of quantum theory. Usually, they are taught in the first weeks of a beginners lecture, and introduced in the ?rst chapters of a textbook. However, their precise operational meaning and a formulation in a general context, i.e. beyond the example of position and momentum observables, are often left out. The reasoning for this is twofold: On one hand, an exact operational definition of uncertainty, indeterminacy and a corresponding uncertainty principle has been the content of many debates since the early days of quantum mechanics until today. From a modern perspective, we have the consent that there are at least the two notions of preparation and measurement uncertainty: the first notion prohibits the existence of dissipation free states and the latter one the existence of error free joint measurements. On the other hand, we have that, the mathematical tools, which are needed for comprehensive treatment of uncertainty relations in a general context, are still under development and usually go far beyond the mathematical level of an introductory course. In this thesis we will investigate these two notions of uncertainty, their corre- sponding uncertainty relations, such as their interplay. The aim of this thesis is to give answers to the central questions: (1.) Which quantities should be used to formulate uncertainty? (2.) How can we compute uncertainty relations for those? (3.) Are there connections between the two notions of uncertainty? We will do this, whenever possible, in a most general context and with a focus on relevant examples, otherwise. Therefore, we will consider constructions of mea- surement errors and deviation measures that quantify uncertainty, based on, so called, cost functions. Commonly used uncertainty measures like variances, en- tropies, and the Hamming distance are examples for these. We will investigate the structure of the corresponding uncertainty relations and provide several methods that enable us to compute them. The third question is addressed by a theorem that shows, for sharp observables, that measurement uncertainty relations can be lower bounded by preparation uncertainty relations, whenever the same cost function is used.