Complex-extended Bohmian mechanics

In Bohm's causal or trajectory interpretation of quantum mechanics, it is straightforward to determine, from a sufficiently large number of calculated particle trajectories, probability distributions for time delays caused by a potential barrier, or for transmission times through a barrier. We show that these distributions can be calculated directly and more efficiently from probability currents, without the calculation of Bohm trajectories as an intermediate step. The ideas are illustrated for Gaussian wave packets incident on a square potential barrier and used to explain why average causal delay times differ from the average delay times calculated in other approaches.

This paper introduces building blocks for modular design of elliptic, pseudoelliptic, and self-equalized filters. The first building block is of second order and generates two transmission zeros (TZs), which are either on the real or imaginary axis. Moving the zeros from the real axis (linear phase response) to the imaginary axis (attenuation poles) requires changing the sign of one coupling coefficient. The second building block is a structure of order three, called extended doublet, which allows the generation of two TZs practically anywhere in the complex plane. An important property of this block is its ability to move the two TZs from the real axis to the imaginary axis of the complex s-plane without changing the signs of its coupling coefficients. The third building block is of third order and generates two TZs, which can be moved from the real to the imaginary axis by changing the sign of one coupling coefficient. Simple waveguide structures to implement these blocks are introduced for validation, although this general approach is feasible for all resonator filter types. Higher order filters are designed modularly by cascading an arbitrary number of these building blocks. A novel concept, which allows the independent control of each pair of TZs in higher order filters, is then introduced. It is shown that the new concept allows a filter of order N to generate N TZs without directly coupling the source to the load even when the coupling coefficients are all assumed frequency independent. The same approach can be used to design and reduce the sensitivity of higher order elliptic and pseudoelliptic filters using other building blocks such as doublets or a mixture of building blocks of different orders and properties. Measured results and extensive computer simulation are presented to demonstrate the validity of the concept and the performance of the designed filters.

We construct a new method for approximating Hilbert transforms and their inverse throughout the complex plane. Both problems can be formulated as Riemann–Hilbert problems via Plemelj’s lemma. Using this framework, we rederive existing approaches for computing Hilbert transforms over the real line and unit interval, with the added benefit that we can compute the Hilbert transform in the complex plane. We then demonstrate the power of this approach by generalizing to the half line. Combining two half lines, we can compute the Hilbert transform of a more general class of functions on the real line than is possible with existing methods.

We have developed a 2.5D finite-difference algorithm to model the elastic wave propagation in heterogeneous media. In 2.5D problems, it is assumed that the elastic properties of models are invariant along a certain direction. Therefore, we can convert the 3D problem into a set of 2D problems in the spectral domain. The 3D solutions are then obtained by applying a numerical integration in the spectral domain. Usually, the quadrature points used in the numerical integration scheme are sampled from the real axis in the spectral domain. The convergence of this quadrature can be very slow especially at high frequencies. We have applied the optimal quadrature scheme for the spectral integration. This is equivalent to transforming the contour of integration from the real axis into a path in the complex plane. Our numerical studies have indicated more than 10 times of reduction in the number of quadrature points compared with sampling along the real axis in the spectral domain. This scheme alleviates the bottleneck of computing speed in the 2.5D elastic wave modeling. Furthermore, it can improve the computational efficiency of 2.5D elastic full-waveform inversion algorithms.

In Bohmian mechanics, the nodes of the wave function play an important role in the generation of chaos. However, so far, most of the attention has been on moving nodes; little is known about the possibility of chaos in the case of stationary nodes. We address this question by considering stationary states, which provide the simplest examples of wave functions with stationary nodes. We provide examples of stationary wave functions for which there is chaos, as demonstrated by numerical computations, for one particle moving in three spatial dimensions and for two and three entangled particles in two dimensions. Our conclusion is that the motion of the nodes is not necessary for the generation of chaos. What is important is the overall complexity of the wave function. That is, if the wave function, or rather its phase, has a complex spatial variation, it will lead to complex Bohmian trajectories and hence to chaos. Another aspect of our work concerns the average Lyapunov exponent, which quantifies the overall amount of chaos. Since it is very hard to evaluate the average Lyapunov exponent analytically, which is often computed numerically, it is useful to have simple quantities that agree well with the average Lyapunov exponent. We investigate possible correlations with quantities such as the participation ratio and different measures of entanglement, for different systems and different families of stationary wave functions. We find that these quantities often tend to correlate to the amount of chaos. However, the correlation is not perfect, because, in particular, these measures do not depend on the form of the basis states used to expand the wave function, while the amount of chaos does.

We describe a series of experimental analogies between fluid mechanics and quantum mechanics recently discovered by a team of physicists. These analogies arise in droplet systems guided by a surface (or pilot) wave. We argue that these experimental facts put ancient theoretical work by Madelung on the analogy between fluid and quantum mechanics into new light. After re-deriving Madelung’s result starting from two basic fluid mechanical equations (the Navier–Stokes equation and the continuity equation), we discuss the relation with the de Broglie–Bohm theory. This allows to make a direct link with the droplet experiments. It is argued that the fluid mechanical interpretation of quantum mechanics, if it can be extended to the general N-particle case, would have a considerable advantage over the Bohm interpretation: it could rid Bohm’s theory of its non-local character.

Semi-classical theories are approximations to quantum theory that treat some degrees of freedom classically and others quantum mechanically. In the usual approach, the quantum degrees of freedom are described by a wave function which evolves according to some Schrödinger equation with a Hamiltonian that depends on the classical degrees of freedom. The classical degrees of freedom satisfy classical equations that depend on the expectation values of quantum operators. In this paper, we study an alternative approach based on Bohmian mechanics. In Bohmian mechanics the quantum system is not only described by the wave function, but also with additional variables such as particle positions or fields. By letting the classical equations of motion depend on these variables, rather than the quantum expectation values, a semi-classical approximation is obtained that is closer to the exact quantum results than the usual approach. We discuss the Bohmian semi-classical approximation in various contexts, such as nonrelativistic quantum mechanics, quantum electrodynamics and quantum gravity. The main motivation comes from quantum gravity. The quest for a quantum theory for gravity is still going on. Therefore a semi-classical approach where gravity is treated classically may be an approximation that already captures some quantum gravitational aspects. The Bohmian semi-classical theories will be derived from the full Bohmian theories. In the case there are gauge symmetries, like in quantum electrodynamics or quantum gravity, special care is required. In order to derive a consistent semi-classical theory it will be necessary to isolate gauge-independent dependent degrees of freedom from gauge degrees of freedom and consider the approximation where some of the former are considered classical.

The mechanical properties of viscoelastic materials are usually described by a rheological model composed of spring and dashpot in series or in parallel or their combination. The complex modulus as a function of angular frequency was firstly deduced, when fractional derivative rheological model is subjected to a sinusoidal perturbation in dynamic mechanical analysis (DMA) measurements. Then, the algebraic equations between the storage and loss moduli of the fractional Zener model in complex plane have been developed. The curve on the complex plane, a plot of loss modulus against storage modulus, is a repressed or distorted semicircle with its center below the real axis. The parameters of mechanical elements can be graphically estimated from its complex plane plot. The dynamic mechanical test of bituminous mixtures was carried out by frequency sweep over a wide range of 200 Hz to 0.02 Hz and its mechanical response can be described adequately by the fractional Zener model, whose element parameters are determined via geometric method.

Bohmian mechanics is arguably the most naively obvious embedding imaginable of Schrödinger's equation into a completely coherent physical theory. It describes a world in which particles move in a highly non-Newtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. It turns out, however, that as a consequence of the defining dynamical equations of Bohmian mechanics, when a system has wave function ψits configuration is typically random, with probability density ρgiven by |ψ|2, the quantum equilibrium distribution. It also turns out that the entire quantum formalism, operators as observables and all the rest, naturally emerges in Bohmian mechanics from the analysis of “measurements.” This analysis reveals the status of operators as observables in the description of quantum phenomena, and facilitates a clear view of the range of applicability of the usual quantum mechanical formulas.

We compare and contrast two distinct approaches to understanding the Born rule in de Broglie-Bohm pilot-wave theory, one based on dynamical relaxation over time (advocated by this author and collaborators) and the other based on typicality of initial conditions (advocated by the 'Bohmian mechanics' school). It is argued that the latter approach is inherently circular and physically misguided. The typicality approach has engendered a deep-seated confusion between contingent and law-like features, leading to misleading claims not only about the Born rule but also about the nature of the wave function. By artificially restricting the theory to equilibrium, the typicality approach has led to further misunderstandings concerning the status of the uncertainty principle, the role of quantum measurement theory, and the kinematics of the theory (including the status of Galilean and Lorentz invariance). The restriction to equilibrium has also made an erroneously-constructed stochastic model of particle creation appear more plausible than it actually is. To avoid needless controversy, we advocate a modest 'empirical approach' to the foundations of statistical mechanics. We argue that the existence or otherwise of quantum nonequilibrium in our world is an empirical question to be settled by experiment.

A method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) —nonlinear ODEs in the complex plane—by generalizing Prelle–Singer has been developed. The approach that the authors generalized is a procedure of obtaining a solution to a kind of second-order nonlinear ODEs in the real line. Some theoretical work has been illustrated and applied to several examples. Also, an extended technique of generating second and third motion integrals in the complex domain has been introduced, which is conceptually an analog to the motion in the real line. Moreover, the procedures of the method mentioned above have been verified.

Bohmian mechanics without wave function ontology

The quantum formalism is a “measurement” formalism-a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from Bohmian mechanics, which is what emerges from Schrödinger's equation for a system of particles when we merely insist that “particles” means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We find that a Bohmian universe, though deterministic, evolves in such a manner that anappearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by “ρ = ¦ψ¦ 2”. A crucial ingredient in our analysis of the origin of this randomness is the notion of the effective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate.

A central issue in hadron spectroscopy is to deduce --- and interpret --- resonance parameters, namely pole positions and residues, from experimental data, for those are the quantities to be compared to lattice QCD or model calculations. However, not every structure in the observables derives from a resonance pole: the origin might as well be branch points, either located on the real axis (when a new channel comprised of stable particles opens) or in the complex plane (when at least one of the intermediate particles is unstable). In this paper we demonstrate first the existence of such branch points in the complex plane and then show on the example of the piN P11 partial wave that it is not possible to distinguish the structures induced by the latter from a true pole signal based on elastic data alone.

The time series plot of electricity daily load demand is seasonal as shown by its regular repetitive pattern during the same period each year. Its behavior is determined by phase-space diagrams that are able to identify any of the following states of the series: fixed point, periodic, or chaotic. The first two deal with predictable systems. This paper focuses on presenting a new methodology to analyze the dynamics of the series in reference by using the curve formed by attractors that move in the complex plane over the Mandelbrot set according to the law dictated by the load curve. Because electrical power is a variable, it is also defined in the complex plane with the components of active power on the real axis and reactive power on the imaginary axis. Therefore, electrical power facilitates a new field of analysis in Mandelbrot fractal space. The obtained temporal curve confirms that the profile of the electric power demand is also mapped with the new fractal geometric space of the Mandelbrot set, thus providing a new contribution that extends knowledge about the dynamics of systems in fractal geometry.