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In this chapter, we review basic concepts from quantum mechanics that will be required for the study of superconducting quantum circuits. We review the fundamental idea of energy quantization and how this can be formalized, using Dirac's ideas, to develop a quantum mechanical description that is consistent with the classical theory for a comparable object. We review the notions of quantum state, observable and projective and generalized measurements, particularizing some of these ideas to the simple case of a two-dimensional object or qubit.
Here we discuss the possible relation of our general conjecture on global attractors of nonlinear Hamiltonian PDEs to dynamical treatment of Bohr's postulates and of wave--particle duality, which are fundamental postulates of quantum mechanics, in the context of coupled nonlinear Maxwell--SchrödingerandMaxwell--Dirac equations. The problem of a dynamical treatment was the main inspiration for our theory of global attractors of nonlinear Hamiltonian PDEs.
The interest taken by Surrealists in alchemy has been well known since the late 1940s, but knowledge of their preoccupation with modern science is more recent. This chapter observes the Surrealist penchant for premodern, occultist epistemologies before focusing on their take up of modern physics in the early 1920s. The theory of relativity (1905 and 1915–16) and developments in quantum mechanics (1922–7) were then undergoing popularization. Apart from popular articles in newspapers/journals, this occurred partly through physicists’ own writings and partly through the philosophy of science. This chapter indicates the importance to Surrealism of the writings of the French philosopher of science, Gaston Bachelard. It also features a case study of the work of German physicist Pascual Jordan whose attempt to extend the findings of quantum mechanics to biology was known to Max Ernst and used by the Surrealists to justify the rejection of positivism. So modern physics became a means of retrospectively comprehending the Surrealists’ turn towards automatism and Ernst’s own natural history incursions. His response to Jordan’s writings offers an alternative way of reading his work.
In addition to his ground-breaking research, Nobel Laureate Steven Weinberg is known for a series of highly praised texts on various aspects of physics, combining exceptional physical insight with his gift for clear exposition. Describing the foundations of modern physics in their historical context and with some new derivations, Weinberg introduces topics ranging from early applications of atomic theory through thermodynamics, statistical mechanics, transport theory, special relativity, quantum mechanics, nuclear physics, and quantum field theory. This volume provides the basis for advanced undergraduate and graduate physics courses as well as being a handy introduction to aspects of modern physics for working scientists.
This chapter examines Cassirer's view on contemporary science. It revisits Cassirer's lesser-known work Determinism and Indeterminism in Modern Physics and argues that it harbors a significantly new stage of his philosophy of physical science. On the one hand, this work presents the quantum formalism as a limiting pole of the Bedeutungsfunktion, the highest mode of symbolic formation according to Cassirer’s “phenomenology of cognition.” Inspired by Paul Dirac, Cassirer understands quantum mechanics as a symbolic calculus for deriving probabilistic predictions of measurement outcomes without regard to underlying wave or particle “images” – or, as an exemplar of abstract symbolic thought. On the other hand, Cassirer recognizes the philosophical significance of the use of group theory in quantum mechanics as advancing a purely structural concept of object in physics. Hence, Ryckman reveals that Cassirer drew epistemological consequences from the symbolic character of contemporary physical theory that retain relevance for philosophy of science today.
In this first capstone chapter we aim to set classical mechanics in context. Classical mechanics played a key role in developing modern physics in the first place, and in turn modern physics has given us deeper insights into the meaning and validity of classical mechanics. Classical mechanics, even extended into the realm of special relativity, has its limitations. It arises as a special case of the vastly more comprehensive theory of quantum mechanics. Where does classical mechanics fall short, and why is it limited? The key to understanding this is Hamilton’s principle. We begin with the behavior of waves in classical physics, and then show results of some critical experiments that upset traditional notions of light as waves and atoms as particles. We proceed to give a brief review of Richard Feynman’s sum-over-paths formulation of quantum mechanics, which describes the actual behavior of light and atoms, and then show that Hamilton’s principle emerges naturally in a certain limiting case.
Chapter 6 covers the internal energy E, which is the first term in the free energy, F = E – TS. The internal energy originates from the quantum mechanics of chemical bonds between atoms. The bond between two atoms in a diatomic molecule is developed first to illustrate concepts of bonding, antibonding, electronegativity, covalency, and ionicity. The translational symmetry of crystals brings a new quantum number, k, for delocalized electrons. This k-vector is used to explain the concept of energy bands by extending the ideas of molecular bonding and antibonding to electron states spread over many atoms. An even simpler model of a gas of free electrons is also developed for electrons in metals. Fermi surfaces of metals are described. The strength of bonding depends on the distance between atoms. The interatomic potential of a chemical bond gives rise to elastic constants that characterize how a bulk material responds to small deformations. Chapter 6 ends with a discussion of the elastic energy generated when a particle of a new phase forms inside a parent phase, and the two phases differ in specific volume.
The new edition of this popular textbook provides a fundamental approach to phase transformations and thermodynamics of materials. Explanations are emphasised at the level of atoms and electrons, and it comprehensively covers the classical topics from classical metallurgy to nanoscience and magnetic phase transitions. The book has three parts, covering the fundamentals of phase transformations, the origins of the Gibbs free energy, and the major phase transformations in materials science. A fourth part on advanced topics is available online. Much of the content from the first edition has been expanded, notably precipitation transformations in solids, heterogeneous nucleation, and energy, entropy and pressure. Three new chapters have been added to cover interactions within microstructures, surfaces, and solidification. Containing over 170 end-of-chapter problems, it is a valuable companion for graduate students and researchers in materials science, engineering, and applied physics.
This chapter extends the discussion of waves beyond the longitudinal oscillations with which we began. Here, we look at the wave equation as it arises in electricity and magnetism, in Euler's equation and its shallow water approximation, in ``realistic" (extensible) strings, and in the quantum mechanical setting, culminating in a quantum mechanical treatment of the book's defining problem, the harmonic oscillator.
We propose a compatibilist theory of free will in the tradition of naturalized philosophy that attempts to: 1) provide a synthesis of a variety of well-known theories, capable of addressing problems of the latter; 2) account for the fact that free will comes in degrees; and 3) interface with neurobiology. We argue that free will comes in degrees, and that these degrees vary with the agent's capacity to make assumptions and use theories. Our model, then, highlights that free-willed actions are consciously monitored by the agent, through beliefs, assumptions, and ultimately theories — hence, the CMT model (for Conscious-through-Monitoring-through-Theories).
The concepts of angular momentum, spin and magnetic moment are worked out using standard quantum mechanical formalism. The concepts of intrinsic spin of a pointlike particle is contrasted with the intrinsic angular momentum of composite particles. The Larmor frequency and the magnetic resonance of non-interacting spins are introduced. The quantum statistics of a system of spins is overviewed, before introducing the thermodynamics of a spin system in a static frame of reference. Nuclear magnetic phase transitions are briefly reviewed.
The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result.
Is the human mind uniquely nonphysical or even spiritual, such that divine intentions can meet physical realities? As scholars in science and religion have spent decades attempting to identify a 'causal joint' between God and the natural world, human consciousness has been often privileged as just such a locus of divine-human interaction. However, this intuitively dualistic move is both out of step with contemporary science and theologically insufficient. By discarding the God-nature model implied by contemporary noninterventionist divine action theories, one is freed up to explore theological and metaphysical alternatives for understanding divine action in the mind. Sarah Lane Ritchie suggests that a theologically robust theistic naturalism offers a more compelling vision of divine action in the mind. By affirming that to be fully natural is to be involved with God's active presence, one may affirm divine action not only in the human mind, but throughout the natural world.
General considerations are given to possible extensions of the methods discussed in the book to nonstationary data beyond 1D signals (e.g., multivariate or graph-based). The analogy between signal theory and quantum mechanics formalisms are mentioned, and a general perpsective on further opportunities is outlined as an open conclusion.
Personal identity is not always symmetric: even if I will not be a later person, the later person may have been me. What makes this possible is that the relations that are criterial of personal identity—such as memory and anticipation—are asymmetric and ‘count in favor of personal identity from one side only’. Asymmetric personal identity can be accommodated by temporal counterpart theory but not by Lewisian overlapping aggregates of person stages. The question of uncertainty in cases of personal fission (and in Everettian quantum mechanics) is also discussed.