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.
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.
Abstract Semiclassical mechanics, which stems from the old quantum theory, has seen a remarkable revival in recent years as a physically intuitive and computationally accurate scheme for the interpretation of modern experiments. The main text concentrates less on the mathematical foundations than on the global influence of the classical phase space structures on the quantum mechanical observables. Further mathematical detail is contained in the appendices. Worked problem sets are included as an aid to the student. One important thread concerns the sharp separation between quantum interference in the classically accessible phase space and exponential decay into the inaccessible shadow regions. Another is the use of catastrophe theory to eliminate singularities at the classical boundaries between the two types of region. The theory goes on to show how different parts of the wavefunction may be stitched together to obtain accurate quantization formulae for a wide variety of tunnelling and curve-crossing problems. A careful account of classical angle–action transformations is extended to include the evaluation of matrix elements. A variety of extensions of the familiar spectroscopic RKR inversion procedure are given. It is also shown how bifurcations in the phase space structure signal changes in the quantum eigenvalue spectrum from, for example, normal mode to local mode behaviour. Other quantization techniques are shown to extend even to weakly chaotic systems. A chapter on Gaussian wavepacket propagation leads up to the currently popular Herman–Kluk technique. The book concludes with chapters on semiclassical aspects of elastic, inelastic and reactive scattering.
Quantum mechanics is our most successful physical theory. It underlies our very detailed understanding of atomic physics, chemistry, and nuclear physics, and the many technologies to which physical systems in these regimes give rise. Additionally, relativistic quantum mechanics is the basis for the standard model of elementary particles, which very successfully gives a partial unification of the forces operating at the atomic, nuclear, and subnuclear levels. However, from its inception the probabilistic nature of quantum mechanics, and the fact that “quantum measurements” in the orthodox formulation appear to require the intervention of non-quantum mechanical “classical systems,” have led to speculations by many physicists, mathematicians, and philosophers of science that quantum mechanics may be incomplete. Among the Founding Fathers of quantum theory, Einstein and Schrödinger were both of the opinion that quantum mechanics is in some way unsatisfactory, and this view has been amplified in more recent profound work of John Bell, among others. In an opposing camp, many others in the physics, mathematics, and philosophy communities have attempted to provide an interpretational foundation in which quantum mechanics remains a complete and self-contained system. Among the Founding Fathers, Bohr, Born, and Heisenberg maintained that quantum mechanics is a complete system, and a number of recent proposals have been made to improve upon or to provide alternatives to their “Copenhagen Interpretation.” The debate continues, and has spawned an enormous literature.
This chapter considers an approximate method of solving the Schrodinger equation in the case when the wavelength of the particle is small by comparison with the distance over which the potential changes appreciably. Since the limit of short wavelength corresponds to the limit of classical mechanics, the quasiclassical approximation allows to explore the relation between classical and quantum mechanics. The chapter discusses how to match the quasiclassical wave function f in the region accessible to classical motion with the wave function in the forbidden region. Since the action function is a minimum along the classical trajectory, the pencil of trajectories adjacent to the classical one gives the minimum phase factor. The chapter examines the problem of the escape of a particle from a potential barrier; this problem reduces to the problem of the spreading of a wave packet which describes a particle which at the initial time was inside the barrier.
Introduction Quantum mechanics is based on the intuition that the essential difference between atomic physics and classical physics is the occurrence of discontinuities (see in particular [1,4,58–63]). Quantum mechanics should thus be considered a direct continuation of the quantum theory founded by Planck, Einstein and Bohr. Bohr in particular stressed repeatedly, already before the birth of quantum mechanics, that the discontinuities must lead to the introduction of new kinematical and mechanical concepts, so that indeed classical mechanics and its corresponding conceptual scheme should be abandoned [1,4]. Quantum mechanics tries to introduce the new concepts through a precise analysis of what is ‘observable in principle’. In fact, this does not mean setting up the principle that a sharp division between ‘observable’ and ‘unobservable’ quantities is possible and necessary. As soon as a conceptual scheme is given, one can infer from the observations to other facts that are actually not observable directly, and the boundary between ‘observable’ and ‘unobservable’ quantities becomes altogether indeterminate. But if the conceptual scheme itself is still unknown, it will be expedient to enquire only about the observations themselves, without drawing conclusions from them, because otherwise wrong concepts and prejudices taken over from before will block the way to recognising the physical relationships [Zusammenhänge].
Fundamental physics contains a family of reciprocal relations that appear in distinct theoretical contexts yet share an identical mathematical structure. The wave propagation relation fλ = c [1], the photon energy relation Eλ = hc [3], the matter-wave relation pλ = h [4], and the quantum of action ET = h [2] each preserve the product of two complementary observable quantities through one or more universal constants. These relations arise independently within classical electrodynamics, photon physics, and quantum mechanics, and their common reciprocal structure is generally attributed to mathematical coincidence rather than an underlying common principle. This work investigates whether these relations can be understood as different representations of a common invariant physical content, connected through a principle of representation preservation. The development proceeds in three stages. First, the general mathematical condition imposed by invariant physical content under changes of representation is derived, establishing that complementary representations must transform reciprocally (Theorem 1). Second, a geometric construction based on synchronized concentric circular representations is introduced, from which a representation length emerges through arc differences. Third, the experimentally established constants c [1, 16] and h [2] are introduced as independent normalization principles governing propagation and action, respectively, connecting the abstract geometric structure to measurable physical quantities. Within this framework, the standard wave relation, the photon energy relation, the de Broglie matter-wave relation, and the quantum of action are recovered as complementary representations of the same invariant physical content (Sect. 9). A representation operator is formally defined and shown to form a one-parameter commutative group of content-preserving transformations (Propositions 1–4, Sect. 7). The representation coordinate is computed for the hydrogen ground state, yielding the fine-structure constant α as the dimensionless representation coordinate of the electronic ground state (Sect. 11). The framework does not replace established physical theories nor derive the numerical values of the fundamental constants. It proposes instead a unified geometric interpretation in which wavelength, frequency, momentum, energy, and time appear as reciprocal representations connected through propagation and action normalization.