The Weyl representation in classical and quantum mechanics

A systematic classification of Feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last 15 years, including, of course, the main contributions since the invention of the path integral by Feynman in 1942. An outline of the general theory is given which will serve as a quick reference for solving path integrals. Explicit formulas for the so-called basic path integrals are presented on which the general scheme to classify and calculate path integrals in quantum mechanics is based.

Canonical transformations to action and angle variables and their representation in quantum mechanics IV. Periodic potentials

Classical statistical average values are generally generalized to average values of quantum mechanics. It is discovered that quantum mechanics is a direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, and the general classical statistical uncertain relation is generally generalized to the quantum uncertainty principle; the two lost conditions in classical uncertain relation and quantum uncertainty principle, respectively, are found. We generally expound the relations among the uncertainty principle, singularity and condensed matter stability, discover that the quantum uncertainty principle prevents the appearance of singularity of the electromagnetic potential between nucleus and electrons, and give the failure conditions of the quantum uncertainty principle. Finally, we discover that the classical limit of quantum mechanics is classical statistical mechanics, the classical statistical mechanics may further be degenerated to classical mechanics and we discover that merely stating that the classical limit of quantum mechanics is classical mechanics is a mistake. As application examples, we deduce both the Schrödinger equation and the state superposition principle, and deduce that there exists a decoherent factor from a general mathematical representation of the state superposition principle; the consistent difficulty between statistical interpretation of quantum mechanics and determinant property of classical mechanics is overcome.

The irreversible process of ‘wave packet collapse’ in quantum measurement contradicts the reversible characteristics of the Schrödinger equation and the entire quantum mechanics. This contradiction is actually the origin of measurement problems in quantum mechanics. How to resolve this contradiction is related to the understanding of some philosophical concepts. The Copenhagen School embraces the reversible/irreversible contradictions in quantum measurement through the principle of complementarity. Both von Neumann's measurement hypothesis and Everett's many world explanation adhere to the ‘universal position of the Schrödinger equation’, but due to their detachment from the thermodynamic mechanism analysis of quantum measurement processes, they fall into the dilemma of subjectivism or possible state materialization in ontology. Decoherence theory is a natural extension of dissipative structure theory in quantum mechanics, which helps to solve quantum measurement problems by combining the study of classical conversion mechanisms between reversibility and irreversibility in statistical mechanics. Due to the common statistical roots of the time arrow problem in classical mechanics and quantum mechanics, decoherence theory cannot be seen as a measurement theory that fully implements the ‘universal Schrödinger position’. Many contradictions in quantum mechanics may need to be cleverly resolved through various contradictions in classical physics.

Poincaré's theorem and unitary transformations for classical and quantum systems

In this paper, I review one of the most important and interesting parts of my new book “Fuzzy Systems to Quantum Mechanics” (see Ref. 1). Several conclusions in this part are worth introducing here. First of all, the motion of a mass point in classic mechanics has also waviness and the wave function of the motion of a mass point is composed of wave functions of countably infinite microscopic particles. Secondly, based on the waviness of the motion of a mass point we surely know the new conclusion described as the wave-mass-point dualism in classic mechanics. And thirdly, by using the closed relation between the wave-mass-point dualism in classic mechanics and the wave-particle dualism in quantum mechanics, unified theory of classic mechanics and quantum mechanics is naturally formed.

In this and the next chapter, we begin a study of the behavior of the spectrum of Schrodinger operators in the semiclassical regime. There is much literature on these topics, and we provide an outline in the Notes to this chapter. The material here follows a part of the work of Simon [Sim5]. In quantum mechanics, the Laplacian plays the role of the energy of a free particle. But, the Laplacian apparently has the dimensions of (length)-2. This is because in our discussions of the Schrodinger operator, we have chosen to work in simple units in which the mass m = 1/2 and Planck’s constant h is taken to be 2π. In these units, the coefficient of the kinetic energy H 0 = -△ is 1. If we restore these constants, the actual differential operator in quantum mechanics representing the energy of a free particle is. The Planck constant h is approximately 6.624 x 10-27 ergs/sec. This is very small except on atomic scales. That is, the length scales over which quantum effects are important depend on h (for example, the Compton wavelength of an electron or the Bohr radius of an atom). This observation provides us with one way to understand the transition from classical to quantum phenomena. We consider quantum theory in which Planck’s constant has been replaced by a small parameter. We then try to understand quantum mechanics in terms of the classical theory obtained as the limit of this quantum theory as the parameter is taken to zero. We call this limit h = 0 the classical limit since all quantum effects have been suppressed. The regime in which this parameter is nonzero, but taken arbitrarily small, is called the semiclassical regime. In this regime, the behavior of the quantum mechanical system should be dominated by the limiting h = 0 classical system.

A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer's information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer's information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process.

The quantum driven pendulum can be implemented using laser cooled rubidium atoms which are exposed to a far detuned amplitude modulated standing wave. This system allows controlled variation of the effective Planck's constant and the amount of coupling to the environment (decoherence). Therefore it is possible to undertake experiments in both classical and quantum regimes of the dynamics of these atoms. The potential can be extremely well approximated as one-dimensional, making the system ideal for simulation using a master equation or quantum trajectories. The dynamics of the system feature distinct regions of regular motion (second order resonances) bounded by a sea of chaos. The resonances are observed as side peaks in the atomic momentum distribution, which each contain up to 30% of the atoms initially in the trap. We report on the results of our experiments to achieve quantum phase space tunneling. After adiabatic elimination of the excited state, the Hamiltonian describing the system is given. The evolution of the quantum driven pendulum can be described with the Floquet operator. Eigenstates of the Floquet operator may be identified with fixed points in phase space. Starting the evolution in a superposition of two Floquet states which is localised in one resonance, time evolution leads to localisation in the other fixed point. This is a coherent tunneling process forbidden by classical dynamics. To observe this tunneling process it is important to localise the superposition of Floquet states in one fixed point. We discuss the effectiveness of alternative methods of loading most of the atoms into a single resonance in order to observe quantum phase space tunneling.

In this article, a quantum analogue of a classical chaotic system is constructed, using the Chen system and a generalized Lorenz system as an example. It is shown that if a classical system in R3 has a chaotic attractor, then a corresponding quantized system can be obtained by the density matrix theory in quantum mechanics. Furthermore, a direct transformation can be used to ensure that the basin of attraction contains the unit Bloch ball with the attractor located in its interior. As an application, a natural model for the Chen qubit is constructed and then extended to the generalized Lorenz qubit. Moreover, the Chen qutrit model is established.

This paper proposes a new method to extract fetal ECG from multi-channel signals. The fetal ECG is extracted in phase space, which is constructed by a pregnant woman's multiple thoracic and abdominal ECG signals. Based on corresponding points in phase space, we adopt least square algorithm to estimate the maternal components of the abdominal signals and extract the fetal signals by subtracting the estimated maternal signals from the abdominal signals. Compared with traditional separation method based on information fusion of multi-channel signals, such as Independent Component Analysis (ICA), this method could be realized more easily and effectively.

Typicality is a well-established and very general property of quantum many-body systems, referring to the phenomenon that the expectation values of any given observable are practically indistinguishable for the overwhelming majority of all pure states (normalized vectors) in a sufficiently high-dimensional Hilbert (sub-)space. Here, we provide very simple and general arguments that analogous typicality properties of pure states (phase space points) in classical many-body systems are still expected to hold true for macroscopic observables, but not any more for microscopic (few-body) observables.

This Element explores the formal and conceptual foundations of phase space formulations of classical and quantum mechanics. It provides an overview of the core mathematical and physical content of Hamiltonian mechanics, stochastic phase space mechanics, contact Hamiltonian mechanics, and open and closed quantum mechanics on phase space. The formal material is unified via three interpretative themes relating to structured possibility spaces, Liouville's theorem and its failure, and the classical and quantum notions of open and closed systems. This Element book is intended for researchers and graduate students in the philosophy and foundations of physics with an interest in the conceptual foundations of physical theory.

Memristors exhibit controllable nonlinear characteristics, generating chaotic signals that are characterized by randomness, sensitivity, and unpredictability, thereby demonstrating significant potential applications in information encryption and signal processing. With the integration of chaos theory and electronic technology, constructing memristive hyperchaotic systems has become a hot topic in nonlinear science and information security. To overcome the limitation of monotonic dynamic characteristics in traditional chaotic systems, we design a novel memristor-based hyperchaotic system with richer dynamic behavior and higher application value in this paper. Moreover, the characteristic analysis, theoretical verification, application exploration, and hardware implementation are conducted to support the engineering applications of the system. Building upon the classical Chen system, this work is innovatively combined with a cubic nonlinear magnetically controlled memristor model as a feedback element. By establishing a mathematical model of the memristor and coupling it with the state equations of the Chen system, we design a four-dimensional memristor-based hyperchaotic system. First, by integrating numerical computation with differential equation theory, a comprehensive mathematical model is established to analyze fundamental properties, such as symmetry and dissipativity, thereby validating the system’s rationality. Second, the system’s dynamical behaviors are analyzed, including attractor phase diagrams, Lyapunov exponents, power spectra, parameter effects, transient dynamics, and coexisting attractors. Simultaneously, variational methods are utilized to analyze unstable periodic orbits within the system. A symbolic coding approach based on orbital characteristics is established to convert orbital information into symbolic sequences, and orbital pruning rules are explored to provide a basis for optimal orbital control. Furthermore, a digital image encryption method is proposed based on this system. Using chaotic sequences as keys, image pixels are scrambled and diffused. The effectiveness of encryption is validated through histogram analysis, correlation analysis, information entropy evaluation, and testing of anti-attack capabilities. Finally, a DSP-based digital circuit hardware platform is constructed to run the system, and the hardware experimental results are compared with software simulation outcomes. These findings reveal that the introduction of memristors induces linearly distributed equilibrium points in phase space, generating hidden attractors that enrich the chaotic behavior of the system. The simulation of dynamic behavior confirms the rich dynamics of this four-dimensional memristor-based hyperchaotic system. The proposed digital image encryption method demonstrates robust security performance. The DSP hardware experiments and software simulations yield highly consistent attractor phase diagrams, validating the correctness and feasibility of the system.

The authors examine several topical subjects, commencing with a general introduction to path integrals in quantum mechanics and the group theoretical backgrounds for path integrals. Applications of harmonic analysis, polar coordinate formulation, various techniques and path integrals of SU(2) and SU(1,1) are discussed. Soluble examples presented include particle-flux system, a pulsed oscillator, magnetic monopole, the Coulomb problems in curved space and others. The second part deals with the SU(2) coherent states and their applications. Construction and generalization of the SU(2) coherent states, formulation of coherent path integrals for spin and unitary spin, and semiclassical quantization are presented. Applications are made to the study of quantum fluctuation, the nonlinear field model and phase holonomy. The final chapters present the theory of the SU(1,1) coherent states and their applications. The radial Coulomb problem, the Morse oscillator, and the large-N approximation are discussed. Applications to problems in quantum optics such as squeezed states, interaction with the squeezed vacuum states, and phase operator formalism are also included. This book is intended as an introduction to the subject as well as a work of reference.