Though this be madness, yet there is method in't. Shakespeare , Hamlet, Act 2 Scene 2 Introduction Semiclassical mechanics can be broadly understood as the theoretical and experimental study of the interconnections between classical and quantum mechanics. More narrowly, it is a field that uses classical quantities to investigate, calculate, and even explain quantum phenomena. Its methods involve an unorthodox blending of quantum and classical ideas, such as a classical trajectory with an associated quantum phase. For these reasons, semiclassical mechanics is often referred to as “putting quantum flesh on classical bones,” where classical mechanics provides the skeletal framework on which quantum quantities are constructed. There are three primary motivations for semiclassical mechanics: First, in many systems of physical interest, a full quantum calculation is cumbersome or even unfeasible. Second, even when a full quantum calculation is within reach, semiclassical methods can often provide intuitive physical insight into a problem, when the quantum solutions are opaque. And, third, semiclassical investigations can lead to the discovery of new physical phenomena that have been overlooked by fully quantum-mechanical approaches. Semiclassical methods are ideally suited for studying physics in the so-called mesoscopic regime, which can roughly be understood as the domain between the classically described macro-world and the quantum mechanically described micro-world. An area in which semiclassical studies have proven to be particularly fruitful is in the subfield of quantum chaos.
Though this be madness, yet there is method in't. Shakespeare , Hamlet, Act 2 Scene 2 Introduction Semiclassical mechanics can be broadly understood as the theoretical and experimental study of the interconnections between classical and quantum mechanics. More narrowly, it is a field that uses classical quantities to investigate, calculate, and even explain quantum phenomena. Its methods involve an unorthodox blending of quantum and classical ideas, such as a classical trajectory with an associated quantum phase. For these reasons, semiclassical mechanics is often referred to as “putting quantum flesh on classical bones,” where classical mechanics provides the skeletal framework on which quantum quantities are constructed. There are three primary motivations for semiclassical mechanics: First, in many systems of physical interest, a full quantum calculation is cumbersome or even unfeasible. Second, even when a full quantum calculation is within reach, semiclassical methods can often provide intuitive physical insight into a problem, when the quantum solutions are opaque. And, third, semiclassical investigations can lead to the discovery of new physical phenomena that have been overlooked by fully quantum-mechanical approaches. Semiclassical methods are ideally suited for studying physics in the so-called mesoscopic regime, which can roughly be understood as the domain between the classically described macro-world and the quantum mechanically described micro-world. An area in which semiclassical studies have proven to be particularly fruitful is in the subfield of quantum chaos.
There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy. Shakespeare , Hamlet, Act 1 Scene 5 The challenge of semiclassical mechanics: Reassessing the views Semiclassical mechanics significantly broadens our understanding of the relationship between classical and quantum mechanics. It reveals that there is much more to the relation between these theories than simply showing that classical behavior can be recovered from quantum systems in the limit of large actions ( ħ → 0) or large quantum numbers ( n → ∞). Although these limits are an important piece of the puzzle, the usual reductionist account of the quantum–classical relation in terms of these limits alone does not tell the whole story. More specifically, research in semiclassical mechanics has uncovered a number of surprising theoretical and experimental correspondences . Indeed the program of semiclassical research can be broadly understood as the theoretical and experimental task of uncovering these correspondences and extending them in new ways. On the theoretical side, semiclassical methods such as EBK quantization, Gutzwiller's trace formula, and closed orbit theory show how classical structures can be used to construct quantum quantities. These semiclassical methods have proven to be not just useful calculational tools, but also important investigative tools, in that they are often able to provide physical insight into the structure of the quantum dynamics, and hence lead to new discoveries.
… there is something in this more than natural, if philosophy could find it out. Shakespeare , Hamlet, Act 2 Scene 2 Introduction The philosophical importance of semiclassical appeals to classical structures in explaining quantum phenomena was first recognized by Robert Batterman (1993; 1995; 2002). More generally, Batterman has argued that when it comes to explaining phenomena in the asymptotic domain between two theories, the fundamental theory describing that domain is often explanatorily deficient, and an adequate explanation must make essential reference to the less fundamental theory. For example, in discussing the relation between classical and quantum mechanics, Batterman writes, “There are many aspects of the semiclassical limit of quantum mechanics that cannot be explained purely in quantum mechanical terms, though they are in some sense quantum mechanical … [T]hese quantum mechanical features require reference to classical properties for their full explanation” (Batterman 2002, pp. 109–10). He has extended these arguments to other theory pairs with singular limits as well, arguing that in optics, for example, one finds explanations of wave-theoretic phenomena that make an essential and ineliminable appeal to ray-theoretic structures such as caustics. More recently, Batterman's arguments have been the subject of considerable criticism from figures such as Clifford Hooker (2004), Michael Redhead (2004), and Gordon Belot (2005). These criticisms have tended to cluster around the following two assumptions: First, in saying that a classical structure (such as a trajectory or caustic) explains , one is thereby committed to the claim that the structure exists .
Abstract Semiclassical mechanics is a short-wavelength link between classical and quantum mechanics, similar to that between wave and geometrical optics. Its object is to maintain contact with the structure imposed by classical mechanics, without sacrificing quantum mechanical accuracy, or, in the vivid words of Berry and Mount (1972), to ‘sew quantum mechanical flesh onto classical bones’. The overriding principles are that the classical phase space is divided into ‘accessible’ and ‘inaccessible’ regions characteristic of the type of motion, and that the boundaries of these regions impose a classical structure, which is evidenced in the quantum mechanics by interference in the bright accessible regions and exponential decay into the shadow regions. The strength of the classical skeleton is that this structure is maintained whether the observations are made in coordinate space, quantum number space, or some mixed representation. In all cases the central quantity is the classical action, whose dimensions are those of Planck’s constant; on one hand it guides the classical motion, according to the minimum action principle, and on the other it adds a phase to each classical trajectory, which gives rise both to quantum mechanical interference and, by a mathematical extension, to quantum mechanical tunnelling.
This paper discusses the problem of finding and defining chaos in quantum mechanics. While chaotic time evolution appears to be ubiquitous in classical mechanics, it is apparently absent in quantum mechanics in part because for a bound, isolated quantum system, the evolution of its state is multiply periodic. This has led a number of investigators to search for semiclassical signatures of chaos. Here I am concerned with the status of semiclassical mechanics as a distinct third theory of the asymptotic domain between classical and quantum mechanics. I discuss in some detail the meaning of such crucial locutions as the "classical counterpart to a quantum system" and a quantum system's "underlying classical motion". A proper elucidation of these concepts requires a semiclassical association between phase space surfaces and wave-functions. This significance of this association is discussed in some detail.
Quantum Chaos provides a comprehensive overview of our understanding of chaotic behaviour in a wide variety of quantum and semiclassical systems, and describes both experimental and theoretical investigations. A general introduction sets out the main features of chaos in quantum systems. Thereafter, in an authoritative collection of papers, prominent scientists put forward their particular interpretations of quantum chaos, with reference to a broad range of interesting physical systems. As yet, there is no universally accepted definition of quantum chaos. However, by dealing with such a wide range of topics from different branches of physics, this book provides an overview of this rapidly expanding field, and will be of great interest to graduate students and researchers in many areas of physics and chemistry.
Physical systems have been traditionally described in terms of either classical or quantum mechanics. But in recent years, semiclassical methods have developed rapidly, providing deep physical insight and computational tools for quantum dynamics and spectroscopy. In this book, Eric Heller introduces and develops this subject, demonstrating its power with many examples. In the first half of the book, Heller covers relevant aspects of classical mechanics, building from them the semiclassical way through the semiclassical limit of the Feynman path integral. The second half of the book applies this approach to various kinds of spectroscopy, such as molecular spectroscopy and electron imaging and quantum dynamical systems with an emphasis on tunneling. Adopting a distinctly time-dependent viewpoint, Heller argues for semiclassical theories from experimental and theoretical vantage points valuable to research in physics and chemistry. Featuring more than two hundred figures, the book provides a geometric, phase-space, and coordinate-space pathway to greater understanding. Filled with practical examples and applications, The Semiclassical Way to Dynamics and Spectroscopy is a comprehensive presentation of the tools necessary to successfully delve into this unique area of quantum mechanics.