Abstract In this article Prof. Ludwig Faddeev sketches a rather formal framework into which Quantum Mechanics satisfactorily fits. He shows that both Classical and Quantum Mechanics can be realized in terms of the same objects. Quantum Mechanics can be understood as a deformation of Classical Mechanics, the mathematical deformation parameter being Planck's constant. (Red.)
Abstract In this article Prof. Ludwig Faddeev sketches a rather formal framework into which Quantum Mechanics satisfactorily fits. He shows that both Classical and Quantum Mechanics can be realized in terms of the same objects. Quantum Mechanics can be understood as a deformation of Classical Mechanics, the mathematical deformation parameter being Planck's constant. (Red.)
This article analyzes the scientific problems that arose at the end of the 19th and beginning of the 20th centuries due to the limitations of classical physics, particularly the issues of measuring the speed of light and the radiation of a blackbody. This article highlights the concept of quantization proposed by M. Planck and its significance in resolving these problems. The formation of quantum mechanics, its main concepts, its role in describing the laws of motion of micro-objects, and its connection with classical mechanics are discussed in detail in this section. Furthermore, this article examines the fundamental concepts of quantum mechanics: discreteness, wave-particle duality, and the significance of Planck’s constant. This article also addresses the challenges of teaching quantum mechanics and ways to overcome them, including the effectiveness of using the case study method.
One can introduce so-called {\em Plain Mechanics} having an {\bf operator realization}. Then the set of one-dimension representations of this operator realization may be identified with the Classical Mechanics. Different irreducible infinite-dimension representations may be recognized as Quantum Mechanics for different ℏ (the Planck constant). It can be done in the such manner that the following diagram will be commutative. Plain Mechanics / \ / \ / \ Quantum Mechanics --> Classical Mechanics h->0 Here the horizontal arrow is well known correspondence between Quantum and Classical Mechanics if Planck constant tensing to zero. A {\em realization} of this scheme for a particle in n-dimensional space by two-sided convolutions on the Heisenberg group is constructed. We also introduce the {\em motion equations} for observables in this realization. The left arrow of the given diagram carries this equation to the Heisenberg one and the right arrow maps it to the Hamilton equation.
Quantum Mechanics and its Interpretations Peter E. Hodgson (bio) Quantum Mechanics is an elegant and indeed beautiful formalism, and enables the results of many physical measurements to be calculated with great accuracy. But from the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything, it is becoming more intense over the years. It should therefore be evident that the most that can be done here is to try to show where the real difficul-ties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. The Discovery of the Quantum The quantum nature of the physical world was discovered by Max Planck in the early years of the twentieth century. He was studying thermodynamics and his attention was drawn to the frequency distribution of black-body radiation. This radiation is emitted from a small hole in a heated enclosure with rough walls, which ensures [End Page 62] that the radiation reaches statistical equilibrium before emission. The frequency distribution is found to be independent of the material lining the walls of the oven, so Planck realized that it must be a fundamental characteristic of all matter. This frequency distribution had been measured by several experimentalists, and Planck resolved to try to see how it can be derived theoretically. Already Rayleigh and Jeans had used classical radiation theory to derive an expression that fitted the data well at low frequencies but tended to infinity at high energies, which was of course unacceptable. The physicist Wien had derived a formula that fitted the data at high frequencies but failed at low frequencies. Planck used an ingenious argument based on entropy to interpolate between these two formulae and obtained a formula, now known as the Planck distribution, that fitted the data perfectly. His next task was to understand the physics behind his formula. He used the familiar technique of assuming that the energy is emitted in small finite amounts, and then let their size tend to zero, so as to represent a continuous distribution. To his great surprise, the intermediate formula with discrete amounts of energy, called quanta, gave the Planck distribution, but the final result, with infinitely small quanta, did not. All his instincts as a physicist rebelled against this conclusion. Surely energy must be emitted continuously. He tried hard to find a way to avoid this unwelcome conclusion, but without success. He had to admit that it is a feature of the world. Soon it was confirmed by Einstein’s interpretation of the photoelectric effect. This story shows clearly that physicists do not impose their ideas on the world, but recognize and accept what they find, whether or not it agrees with their original ideas. The world is made by God in ways that often we cannot imagine. [End Page 63] The Old Quantum Theory A few years later Rutherford discovered that the atom has a small central nucleus containing most of its mass surrounded by a number of electrons. Bohr used the idea of the quantum to build his model of the atom. He assumed that the electrons go around the nucleus in discrete orbits, and that radiation is emitted when an electron falls from one orbit to a lower one. With this model, he was able to calculate to high accuracy the frequencies of the radiation emitted from an excited hydrogen atom. This was a great achievement, but to get his result he had to assume that the electrons stayed in their orbits, whereas classical theory predicted that they would continuously lose energy and rapidly spiral into the nucleus. Quantum Mechanics and its Interpretations These problems were solved by the development of quantum mechanics in the 1920s by Heisenberg, Schrodinger, Dirac, and others. This enabled what is called the wave function to be calculated for any physical system, and from the wave function the probabilities of occurrence of measurable quantities can be found. It is a beautiful theory, and is the indispensable means of calculation for atomic, nuclear...
Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.
Introduction As the decades of the twentieth century rolled on, quantum mechanics (QM) became more and more sophisticated and mathematical. Understanding what this theory means intuitively was confounded not only by nonclassical concepts such as wave–particle duality and quantum interference, but also by issues to do with observation. The Newtonian classical mechanics (CM) paradigm of reality, wherein reductionist laws of physics describe observer-independent dynamics of systems under observation (SUOs) with observer-independent properties, was found to be inadequate. Quantum theorists were confronted with the measurement problem , which attempts to understand, explain, and rationalize the laws of QM that underpin the processes of observation that go on in the laboratory. They are not the same as those of CM in several puzzling respects. Historically, the first sign of the measurement problem was Planck's quantization of energy (Planck, 1900b) and the second was Bohr's veto on radiation damping in hydrogen (Bohr, 1913). These occurred in the first quarter of the twentieth century, a period in physics often referred to as Old Quantum Mechanics . Another indicator that intuition was inadequate was Born's interpretation of the squared modulus of the Schrödinger wave function as a probability density (Born, 1926). That interpretation has everything to do with observers and observation, because probability without an observer is a vacuous concept. Eventually, the projection-valued measure (PVM) formalism emerged, championed by von Neumann in an influential book on the mathematical formulation of QM (von Neumann, 1955). Subsequently, pioneers such as Ludwig (Ludwig, 1983a,b) and Kraus (Kraus, 1974, 1983) refined the theory into the general positive operator-valued measure (POVM) formalism that we shall discuss and use. The quantized detector network (QDN) approach to quantum experiments is most naturally expressed in the PVM formalism, as QDN focuses on the individual detectors in the laboratory. However, the POVM formalism is more general than the PVM formalism, giving a description of multiple detector processes similar to QDN. This raises the question of how the two approaches, QDN and POVM, are related. The aim of this chapter is to explore this relationship. Before we review the PVM and POVM formalisms, we review some essential mathematical concepts.