Semiclassical work and quantum work identities in Weyl representation

Semiclassical quantization of complex Henon–Heiles systems

Official journal of the Institute of Physics - Sri Lanka. Full text articles are available. The journal also has a website at http://www.ip-sl.org/sljp/The new email for the Editor is eic.ipsl@gmail.com.

The Weyl representation in classical and quantum mechanics

With a view to increasing the scope of applications of supersymmetric quantum mechanics (SUSY QM), we formulate the same in a complex phase space. Within this framework, the concept of shape invariance is reinvestigated and an insight into the eigenvalue spectra of non-Hermitian Hamiltonians is sought. The results are applied to a variety of potentials. We claim that the shape invariance for these potentials in the complex phase space can be retrieved in terms of the prescriptions already proposed in the conventional SUSY QM, in that the transformation of potential parameters takes the form of a reflection in the parameter space. Interestingly, some of these features turn out to be the generalization of the concept of quasi-parity used recently in the context of SUSY QM of PT-symmetric potentials. Further, a correspondence of the present approach with other complex formulations of SUSY QM and also in two real dimensions is demonstrated.

In the present work, using the generalized Boltzmann equation of the first author [Phys. Rev. E 94, 023316 (2016)], a quantum kinetic model in the complex phase space is proposed. Employing the Chapman-Enskog analysis and applying the Wick rotation (using complex-valued relaxation time), it is shown that the present model recovers the time-dependent Schr\oedinger equation, while preserving the main features of the conventional lattice Boltzmann models, e.g., simplicity of implementation, second-order accuracy (in space), and convenience in parallel programming. The present results are numerically verified by simulating three benchmark problems.

A finite Hilbert space can be associated to a periodic phase space, that is, a torus. Then a subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure state Wigner and chord functions, involving, for instance the inverse phase space participation ratio and correlations of a state with its translate. New identities are derived involving transition Wigner or chord functions of transition operators . Extension of the reflection and translation operators to a doubled torus phase space leads to the representation of superoperators and so to the construction of the propagator of Wigner functions from the Weyl representation of the evolution operator.

We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.

We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.

Abstract. In recent years, several climate subsystems have been identified that may undergo a relatively rapid transition compared to the changes in their forcing. Such transitions are rare events in general, and simulating long-enough trajectories in order to gather sufficient data to determine transition statistics would be too expensive. Conversely, rare events algorithms like TAMS (trajectory-adaptive multilevel sampling) encourage the transition while keeping track of the model statistics. However, this algorithm relies on a score function whose choice is crucial to ensure its efficiency. The optimal score function, called the committor function, is in practice very difficult to compute. In this paper, we compare different data-based methods (analog Markov chains, neural networks, reservoir computing, dynamical Galerkin approximation) to estimate the committor from trajectory data. We apply these methods on two models of the Atlantic Ocean circulation featuring very different dynamical behavior. We compare these methods in terms of two measures, evaluating how close the estimate is from the true committor and in terms of the computational time. We find that all methods are able to extract information from the data in order to provide a good estimate of the committor. Analog Markov Chains provide a very reliable estimate of the true committor in simple models but prove not so robust when applied to systems with a more complex phase space. Neural network methods clearly stand out by their relatively low testing time, and their training time scales more favorably with the complexity of the model than the other methods. In particular, feedforward neural networks consistently achieve the best performance when trained with enough data, making this method promising for committor estimation in sophisticated climate models.

The method of parametric decomposition is focused on exact methods of the perturbed nonlinear systems' study. The method allows research into the behavior of many-dimensional nonlinear systems through the study of the behavior of the basic subsystems, no higher than the 2nd order, performing rigorous analysis. Investigation of the basic subsystems is proposed using the point transformation method in two-sided phase plane and into complex phase space.

The tennis racket effect is a geometric phenomenon which occurs in a free rotation of a three-dimensional rigid body. In a complex phase space, we show that this effect originates from a pole of a Riemann surface and can be viewed as a result of the Picard-Lefschetz formula. We prove that a perfect twist of the racket is achieved in the limit of an ideal asymmetric object. We give upper and lower bounds to the twist defect for any rigid body, which reveals the robustness of the effect. A similar approach describes the Dzhanibekov effect in which a wing nut, spinning around its central axis, suddenly makes a half-turn flip around a perpendicular axis and the monster flip, an almost impossible skateboard trick.

Highly excited atoms in external fields arc fundamental quantum systems which possess in many cases a classically chaotic analogue. The analysis of their quantum spectra is of fundamental importance for a deeper understanding of phenomena related to the so-called quantum chaos. The development and application of closed-orbit theory was the decisive step for a semiclassical calculation and interpretation of photoabsorption spectra. In its original version formulated by Du and Delos (1988) and Bogomolny (1989) closed-orbit theory was able to reproduce most structures in hydrogenie and even nonhydrogenic atoms but suffered from singularities at energies where orbits undergo bifurcations. Unphysical singularities at bifurcation energies of closed orbits are now removed in uniform semiclassical approximations and the analysis of these uniform solutions also reveals the role and physical importance of "ghost" orbits in complex phase space.

A number of geometric questions related to the complex theory of the canonical Maslov operator are considered. The investigations center around the following themes: 1) quantization conditions and the problem of finding the asymptotics of eigenvalues; 2) universal characteristics of the theory of the complex canonical operator; 3) complex Hamiltonian fields and their trajectories.

Saddle–node bifurcations have been described in a multitude of nonlinear dynamical systems modeling physical, chemical, as well as biological systems. Typically, this type of bifurcation involves the transition of a given set of fixed points from the real to the complex phase space. After the bifurcation, a saddle remnant can continue influencing the flows and generically, for non-degenerate saddle–node bifurcations, the time the flows spend in the bottleneck region of the ghost follows the inverse square root scaling law. Here we analytically derive this scaling law for a general one-dimensional, analytical, autonomous dynamical system undergoing a not necessarily non-degenerate saddle–node bifurcation, in terms of the degree of degeneracy by using complex variable techniques. We then compare the analytic calculations with a one-dimensional equation modeling the dynamics of an autocatalytic replicator. The numerical results are in agreement with the analytical solution.

Kelley’s Geometrical Optics method opens an efficient approach to the study of waves when the variables in the wave equation cannot be separated. The trajectories of the Hamilton-Jacobi equation are then needed for access to the wave structure. When several disjoint potential wells are present, account of the interaction between them requires consideration of trajectories and tori in complex phase space and of the unfamiliar connectivity of such complex toroids. It is shown how connection theory across wells and barriers can be reformulated in a manner that is simple and flexible and assures the rigorous reliability needed for exponentially precise approximation to functionals of the Hamiltonian operator in several dimensions. To make this difficult and unfamiliar subject more accessible, its discussion is preceded by a simple example of a double well on a strip domain in two dimensions. Parts II and III will apply the results to the quantitative resolution of degeneracy of discrete and continuous spectra, respectively, for genuine partial differential equations.