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Quantum Mechanics and its Interpretations

Peter Edward Hodgson

Published Jan 18, 2009• 0 citations
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Abstract

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...

DOI
10.1353/log.0.0027

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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...
Toward the end of the nineteenth century, it became clear that some experiments cannot be explained on the basis of known physics, now referred to as classical physics. In 1900, Max Planck was able to correctly describe the distribution of blackbody radiation energy as a function of wavelength λ or frequency ν = c/λ. He had to accept the existence of energy packets (quanta) with energy E = hν. A few years later, Albert Einstein found that this unexpected relation between energy and frequency is also necessary in order to understand photoemission from metallic surfaces. A decade later, Niels Bohr arrived at the correct equation for the electronic spectrum of the hydrogen atom by applying “quantum conditions” on the motion of the electron around the nucleus and the Planck’s relation E = hν on the energy difference between the energy levels (1916). A decade later, Heisenberg, Born, and Jordan introduced “quantum mechanics” and de Broglie and Schrödinger (1925-1926) introduced “wave mechanics.”
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. 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. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties 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. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
In this chapter, we present the basic principles of the quantum mechanics of particle systems from a modern mathematical perspective. The proper study of quantum mechanics brings together several interesting mathematical ideas, ranging from group representations to the spectral theory of unbounded self-adjoint operators in Hilbert spaces and the theory of C * algebras. The facts from analysis of operators in Hilbert spaces and C * algebras needed in this chapter are presented in detail in the appendices at the end of the book. More on the mathematical foundations of quantum mechanics can be found in the books by L. Fadeev and O. Yakubovskii [FY], G. Mackey [M], and F. Strocchi [Str] listed in the bibliography. The birth of quantum theory The origins of quantum theory can be traced back to M. Planck's study, published in 1900, of so-called black-body radiation. It is well known that matter glows when heated, going from red-hot to white-hot as the temperature rises. The color (or frequency of radiation) of the radiating body is independent of its surface, and for a black body it is found to be a function of temperature alone. The behavior of the radiation energy as a function of temperature seemed quite different at high temperatures and low temperatures. In a desperate attempt to fit these different behaviors into a single law, Planck introduced the hypothesis that energy was not continuous, but came in discrete packets or quanta . At the beginning of the twentieth century, a number of experiments pointed to a breakdown of the laws of classical mechanics at very small (subatomic) scales.
This chapter presents the basic physical ideas that underpin the quantum theory. Historians place the beginnings of the quantum theory at 1900. That year Max Planck published his famous formula for the distribution of energy as a function of frequency in black-body radiation. Curiously, although radiation theory gave birth to quantum mechanics, the interaction of matter and electromagnetic radiation is a rather advanced topic in quantum mechanics and it took almost fifty years for an acceptable formulation to be achieved. One reason for this is that photons are quanta of the electromagnetic field, and the quantum theory of fields requires a relativistic treatment to be entirely successful. The corpuscular nature of light was only half of the input that gave birth to quantum mechanics. The other half concerns matter. Given that the discreteness of photons manifests itself during the emission and absorption process, it is not surprising to find that quantum oddities afflict the behaviour of material particles too.
Newtonian mechanics is the foundation of Classical Physics. Newton’s mechanics, Thermodynamics, Wave theory of Optics and Maxwell’s electromagnetic theory belong to regime of Classical Physics and can be used to explain a wide range of phenomena in the Universe at macroscopic scale. These theories fail spectacularly when applied to phenomenain the atomic and nuclear regime, for example, proton-atom scattering or the flow of electrons in a semiconductor. Quantum mechanics is the most successful scientific theory that has ever been created and it has completely changed our view of the world. The failure of Classical Physics was highlighted by black body radiation and photoelectric effect. Max Planck and Albert Einstein provided explanations of both phenomena based on quantum hypothesis and are thus considered founders of Quantum Physics. The origin of quantum mechanics goes back to the mid-1920s.It was formulated first as matrix mechanics by Werner Heisenberg, Max Born and Pascual Jordan; then as wave mechanics by Louis de Broglie and Erwin Schrödinger; and later on asquantum statistics of subatomic particles by Fermi-Dirac and Bose-Einstein. Combining relativistic mechanics with quantum mechanics, Dirac formulated his relativistic quantum mechanics during 1930s. Uncertainty Principle is thecornerstone of Quantum Physics. The role of randomness in microscopic physical processes shatters the myth that the universe is deterministic. Quantum world is unpredictable in the classical sense and demolishes the idea of an objective universe. The Copenhagen interpretation remains the quantum mechanical formalism that is currently most widely accepted amongst physicists.Quantum theories support cosmic spirit pervading the cosmos and inter – relationship of individuals in the world society. Quantum philosophy is holistic and is going to revolutionize our world-view.
Quantum mechanics began in very early twentieth century when Max Planck found the expression for black body thermal radiation in which the emitted light was not a continuum as postulated by classical physics. Erwin Schrodinger applied his equation to solve the quantum harmonic oscillator, the quantum rigid rotor, and a hydrogen-like atom. Pertaining to the solution of the Schrodinger equation to hydrogen atoms are the quantum numbers. C. K. Ingold’s overview of Schrodinger wave mechanics: “What can be known about an electron, considered as a particle, is of a statistical nature, and is summarized in the behavior of a wave. When considering singlet substances, the normal Zeeman Effect occurs, providing the discrete values of the orbital angular momentum. In an attempt to rationalize the anomalous Zeeman effect, W. Pauli proposed the fourth degree of freedom of the electron which he called “two-valuedness not describable classically”.