Case Of Harmonic Oscillator Research Articles

Optical ladder operators in the Glauber-Fock oscillator array

In this study, we investigate the stationary states of the Glauber-Fock oscillator waveguide array. We begin by transforming the associated Hamiltonian into the form of a quantum harmonic oscillator Hamiltonian, allowing the implementation of a supersymmetric (SUSY) approach. By considering the simplest case for the intertwining operator, the optical ladder operators are straightforwardly constructed and shown to map eigensolutions into eigensolutions of the corresponding Hamiltonian operator, in pretty much the same manner as it is done for the quantum harmonic oscillator case. The ladder of the corresponding (eigen) supermodes is then easily established.

ANALYSIS OF MECHANICAL POWER TRANSFORMATION PROCESSES DURING HARMONIC OSCILLATIONS OF DRIVES IN LOAD-LIFTING, TRANSPORTING AND VIBROFORMING CONCRETE MIXTURE MACHINES AND MECHANISMS

The purpose of the study is to detail the types of mechanical power during harmonic oscillations of drives in load-lifting, transporting and vibroforming concrete mixture machines and mechanisms. Due to the irreversibility of thermal energy, its derivative will take only positive values. However, derivatives can be taken from both potential and kinetic energy. But the most interesting case is the case of harmonic os-cillations in the specified machines and mechanisms drives, for which the derivatives (instantaneous power) are sign-changing functions, that fundamently distinguishes them from thermal power. The analogue of kinetic energy in electric engineering is magnetic field energy in inductor coil, potential energy analogue is capacitor’s electric field energy, and mechanical thermal energy is replaced by thermal energy, which is dissipated by the resistor. It is shown in this study that not only thermal power (with a positive sign) develops during load-lifting, transporting and vibroforming concrete mixture machines and mechanisms drivers mechanical oscillations , but also sign-changing reactive powers, which characterize the kinetic and potential energies inverse. Here, the active power is understood as the average value of the instantaneous power for half a period, and under reactive − amplitude value. The full mechanic power of machines and mechanisms drives, on the one hand, is described by the Pythagorean formula, and from the other − is equal to the product of the harmonic quantities effec-tive values. A feature of the complex representation is that when calculating the full power, one of the vectors that are multiplied must be conjugate. The representation about mechanical reactive, active and full power is a generalization of the corre-sponding consepts from electrical engineering, which is a manifestation of electro-mechanical dualism. Since in most machines and mechanisms the drives are mainly electromechanical, mechanical reactive power transforms into electrical reactive power of the network, thereby worsening the quality of electricity. Therefore, con-sideration of mechanical reactive power is of great importance.

Classical and quantum dynamics of over-damped non linear systems

Overdamping is a regime in which friction is sufficiently large that the motion either decays to its equilibrium position or it crosses the equilibrium position exactly once before returning monotonically towards the equilibrium position. The phenomena of overdamping has been studied classically and quantum mechanically only for the case of the linear damped harmonic oscillator. Here we study the classical and quantum dynamics of a family of over-damped non linear systems. The main objective of this paper is to find a Lagrangian and Hamiltonian framework to study over-damped non linear systems and to show that a quantum mechanical description can be developed in the momentum representation. Our results reduce to the well known solution of the linear damped harmonic oscillator when the non linear part is set to zero.

Superoscillating Sequences and Supershifts for Families of Generalized Functions

We construct a large class of superoscillating sequences, more generally of {mathscr {F}}-supershifts, where {mathscr {F}} is a family of smooth functions in (t, x) (resp. distributions in (t, x), or hyperfunctions in x depending on the parameter t) indexed by lambda in {mathbb {R}}. The frame in which we introduce such families is that of the evolution through Schrödinger equation (ipartial /partial t - {mathscr {H}}(x))(psi )=0 ({mathscr {H}}(x) = -(partial ^2/partial x^2)/2 + V(x)), V being a suitable potential). If {mathscr {F}}= {(t,x) mapsto varphi _lambda (t,x),;, lambda in {mathbb {R}}}, where varphi _lambda is evolved from the initial datum xmapsto e^{ilambda x}, {mathscr {F}}-supershifts will be of the form {sum _{j=0}^N C_j(N,a) varphi _{1-2j/N}}_{Nge 1} for ain {mathbb {R}}{setminus }[-1,1], taking C_j(N,a) =left( {begin{array}{c}N jend{array}}right) (1+a)^{N-j}(1-a)^j/2^N. Our results rely on the fact that integral operators of the Fresnel type govern, as in optical diffraction, the evolution through the Schrödinger equation, such operators acting continuously on the weighted algebra of entire functions mathrm{Exp}({mathbb {C}}). Analyzing in particular the quantum harmonic oscillator case forces us, in order to take into account singularities of the evolved datum that occur when the stationary phasis in the Fresnel operator vanishes, to enlarge the notion of {mathscr {F}}-supershift, {mathscr {F}} being a family of C^infty functions or distributions in (t, x), to that where {mathscr {F}} is a family of hyperfunctions in x, depending on t as a parameter.

Four-body (an)harmonic oscillator in d-dimensional space: S-states, (quasi)-exact-solvability, hidden algebra sl (7)

As a generalization and extension of our previous paper [Turbiner et al., J. Phys. A: Math. Theor. 53, 055302 (2020)], in this work, we study a quantum four-body system in Rd (d ≥ 3) with quadratic and sextic pairwise potentials in the relative distances, rij ≡ |ri − rj|, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum (S-states). In variables ρij≡rij2, the corresponding reduced Hamiltonian of the system possesses a hidden sl(7; R) Lie algebra structure. In the ρ-representation, it is shown that the four-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly solvable. We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite and three others are equal), molecular two-center (two masses are infinite and two others are equal), and molecular three-center (three infinite masses) cases. In particular, exact results in the molecular case are compared with those obtained within the Born–Oppenheimer approximation. The first and second order symmetries of non-interacting system are searched. In addition, the reduction to the lower dimensional cases d = 1, 2 is discussed. It is shown that for the four-body harmonic oscillator case, there exists an infinite family of eigenfunctions that depend on the single variable, which is the moment of inertia of the system.

The Generalized OTOC from Supersymmetric Quantum Mechanics—Study of Random Fluctuations from Eigenstate Representation of Correlation Functions

The concept of the out-of-time-ordered correlation (OTOC) function is treated as a very strong theoretical probe of quantum randomness, using which one can study both chaotic and non-chaotic phenomena in the context of quantum statistical mechanics. In this paper, we define a general class of OTOC, which can perfectly capture quantum randomness phenomena in a better way. Further, we demonstrate an equivalent formalism of computation using a general time-independent Hamiltonian having well-defined eigenstate representation for integrable Supersymmetric quantum systems. We found that one needs to consider two new correlators apart from the usual one to have a complete quantum description. To visualize the impact of the given formalism, we consider the two well-known models, viz. Harmonic Oscillator and one-dimensional potential well within the framework of Supersymmetry. For the Harmonic Oscillator case, we obtain similar periodic time dependence but dissimilar parameter dependences compared to the results obtained from both microcanonical and canonical ensembles in quantum mechanics without Supersymmetry. On the other hand, for the One-Dimensional PotentialWell problem, we found significantly different time scales and the other parameter dependence compared to the results obtained from non-Supersymmetric quantum mechanics. Finally, to establish the consistency of the prescribed formalism in the classical limit, we demonstrate the phase space averaged version of the classical version of OTOCs from a model-independent Hamiltonian, along with the previously mentioned well-cited models.

The Noether–Bessel-Hagen symmetry approach for dynamical systems

The Noether–Bessel-Hagen theorem can be considered a natural extension of Noether Theorem to search for symmetries. Here, we develop the approach for dynamical systems introducing the basic foundations of the method. Specifically, we establish the Noether–Bessel-Hagen analysis of mechanical systems where external forces are present. In the second part of the paper, the approach is adopted to select symmetries for a given systems. In particular, we focus on the case of harmonic oscillator as a testbed for the theory, and on a cosmological system derived from scalar–tensor gravity with unknown scalar-field potential [Formula: see text]. We show that the shape of potential is selected by the presence of symmetries. The approach results particularly useful as soon as the Lagrangian of a given system is not immediately identifiable or it is not a Lagrangian system.

Semiclassical work and quantum work identities in Weyl representation

We derive a semiclassical nonequilibrium work identity by applying the Wigner–Weyl quantization scheme to the Jarzynski identity for a classical Hamiltonian. This allows us to extend the concept of work to the leading order in ℏ. We propose a geometric interpretation of this semiclassical Jarzynski relation in terms of trajectories in a complex phase space and illustrate it with the exactly solvable case of the quantum harmonic oscillator.

Path integral for new models of time-dependent harmonic oscillators

We have presented a particular direct method to solve time-dependent quantum problems within the framework of path integrals by using explicitly time-dependent transformations. We have applied it to the case of harmonic oscillator with time-dependent mass and frequency. Some examples have illustrated the method used. From their exact propagators, normalized solutions have been obtained.

Confinement of two-body systems and calculations in d dimensions

A continuous transition for a system moving in a three-dimensional (3D) space to moving in a lower-dimensional space, 2D or 1D, can be made by means of an external squeezing potential. A squeeze along one direction gives rise to a 3D to 2D transition, whereas a simultaneous squeeze along two directions produces a 3D to 1D transition, without going through an intermediate 2D configuration. In the same way, for a system moving in a 2D space, a squeezing potential along one direction produces a 2D to 1D transition. In this work we investigate the equivalence between this kind of confinement procedure and calculations without an external field, but where the dimension $d$ is taken as a parameter that changes continuously from $d=3$ to $d=1$. The practical case of an external harmonic oscillator squeezing potential acting on a two-body system is investigated in details. For the three transitions considered, 3D~$\rightarrow$~2D, 2D~$\rightarrow$~1D, and 3D~$\rightarrow$~1D, a universal connection between the harmonic oscillator parameter and the dimension $d$ is found. This relation is well established for infinitely large 3D scattering lengths of the two-body potential for 3D~$\rightarrow$~2D and 3D~$\rightarrow$~1D transitions, and for infinitely large 2D scattering length for the 2D~$\rightarrow$~1D case. For finite scattering lengths size corrections must be applied. The traditional wave functions for external squeezing potentials are shown to be uniquely related with the wave functions for specific non-integer dimension parameters, $d$.

Self-consistent time-dependent harmonic approximation for the sine-Gordon model out of equilibrium

We derive a self-consistent time-dependent harmonic approximation for the quantum sine-Gordon model out of equilibrium and apply the method to the dynamics of tunnel-coupled one-dimensional Bose gases. We determine the time evolution of experimentally relevant observables and in particular derive results for the probability distribution of subsystem phase fluctuations. We investigate the regime of validity of the approximation by applying it to the simpler case of a nonlinear harmonic oscillator, for which numerically exact results are available. We complement our self-consistent harmonic approximation by exact results at the free fermion point of the sine-Gordon model.

Bargmann-Type Transforms and Modified Harmonic Oscillators

We study some complete orthonormal systems on the real line. These systems are determined by Bargmann-type transforms, which are Fourier integral operators with complex-valued quadratic phase functions. Each system consists of eigenfunctions for a second-order elliptic differential operator like the Hamiltonian of the harmonic oscillator. We also study the commutative case of a certain class of systems of second-order differential operators called the non-commutative harmonic oscillators. By using the diagonalization technique, we compute the eigenvalues and eigenfunctions for the commutative case of the non-commutative harmonic oscillators. Finally, we study a family of functions associated with an ellipse in the phase plane. We show that the family is a complete orthogonal system on the real line.

Axiomatic construction of quantum Langevin equations

A phenomenological construction of quantum Langevin equations, based on the physical criteria of (i) the canonical equal-time commutators, (ii) the Kubo formula, (iii) the virial theorem and (iv) the quantum fluctuation–dissipation theorem is presented. The case of a single harmonic oscillator coupled to a large external bath is analysed in detail. This allows to distinguish a Markovian semi-classical approach, due to Bedeaux and Mazur, from a non-Markovian full quantum approach, due to to Ford, Kac and Mazur. The quantum-fluctuation–dissipation theorem is seen to be incompatible with a Markovian dynamics. Possible applications to the quantum spherical model are discussed.

The ‘Miracle’ of Applicability? The Curious Case of the Simple Harmonic Oscillator

The paper discusses to what extent the conceptual issues involved in solving the simple harmonic oscillator model fit Wigner’s famous point that the applicability of mathematics borders on the miraculous. We argue that although there is ultimately nothing mysterious here, as is to be expected, a careful demonstration that this is so involves unexpected difficulties. Consequently, through the lens of this simple case we derive some insight into what is responsible for the appearance of mystery in more sophisticated examples of the Wigner problem.

More on homological supersymmetric quantum mechanics

In this work, we first solve complex Morse flow equations for the simplest case of a bosonic harmonic oscillator to discuss localization in the context of Picard-Lefschetz theory. We briefly touch on the exact non-BPS solutions of the bosonized supersymmetric quantum mechanics on algebraic geometric grounds and report that their complex phases can be accessed through the cohomology of WKB 1-form of the underlying singular spectral curve subject to necessary cohomological corrections for non-zero genus. Motivated by Picard-Lefschetz theory, we write down a general formula for the index of $\mathcal{N} = 4$ quantum mechanics with background $R$-symmetry gauge fields. We conjecture that certain symmetries of the refined Witten index and singularities of the moduli space may be used to determine the correct intersection coefficients. A few examples, where this conjecture holds, are shown in both linear and closed quivers with rank-one quiver gauge groups. The $R$-anomaly removal along the "Morsified" relative homology cycles also called "Lefschetz thimbles" is shown to lead to the appearance of Stokes lines. We show that the Fayet-Iliopoulos (FI) parameters appear in the intersection coefficients for the relative homology of the quiver quantum mechanics resulting from dimensional reduction of $2d$ $\mathcal{N}=(2,2)$ gauge theory on a circle and explicitly calculate integrals along the Lefschetz thimbles in $\mathcal{N}=4$ $\mathbb{CP}^{k-1}$ model. The Stokes jumping of coefficients and its relation to wall crossing phenomena is briefly discussed. We also find that the notion of "on-the-wall" index is related to the invariant Lefschetz thimbles under Stokes phenomena. An implication of the Lefschetz thimbles in constructing knots from quiver quantum mechanics is indicated.

Data Rate for Distributed Consensus of Multiagent Systems With High-Order Oscillator Dynamics

Distributed consensus under data rate constraint is an important research topic of multiagent systems. Some results have been obtained for consensus of multiagent systems with integrator dynamics, but it remains challenging for general high-order systems, especially in the presence of unmeasurable states. In this paper, we study the quantized consensus problem for a special kind of high-order critical systems and investigate the corresponding data rate required for achieving consensus. The state matrix of each agent is a 2 mth order real Jordan block admitting m identical pairs of conjugate poles on the unit circle; each agent has a single input, and only the first state variable can be measured. The case of harmonic oscillators (m = 1) is first investigated under a directed communication topology which contains a spanning tree, while the general case of m ≥ 2 is considered for a connected and undirected network. In both cases, the sufficient number of communication bits to guarantee the exponentially fast consensus is shown to be an integer between m and 2 m, depending on the location of the poles.

The Simpson’s paradox in quantum mechanics

In probability and statistics, the Simpson’s paradox is a paradox in which a trend that appears in different groups of data disappears when these groups are combined, while the reverse trend appears for the aggregate data. In this paper, we give some results about the occurrence of the Simpson’s paradox in quantum mechanics. In particular, we prove that the Simpson’s paradox occurs for solutions of the quantum harmonic oscillator both in the stationary case and in the non-stationary case. In the non-stationary case, the Simpson’s paradox is persistent: if it occurs at any time t=t∼, then it occurs at any time t≠t∼. Moreover, we prove that the Simpson’s paradox is not an isolated phenomenon, namely, that, close to initial data for which it occurs, there are lots of initial data (a open neighborhood), for which it still occurs. Differently from the case of the quantum harmonic oscillator, we also prove that the paradox appears (asymptotically) in the context of the nonlinear Schrödinger equation but at intermittent times.

Braid group representations from a deformation of the harmonic oscillator algebra

We describe a new technique to obtain representations of the braid group Bn from the ℜ–matrix of a quantum deformed algebra of the one dimensional harmonic oscillator. We consider the action of the ℜ–matrix not on the tensor product of representations of the algebra, that in the harmonic oscillator case are infinite dimensional, but on the subspace of the tensor product corresponding to the lowest weight vectors.

Space–time transformation for the propagator in de Broglie–Bohm theory

A linear space–time transformation proposed to calculate the propagator in the de Broglie–Bohm theory, viewed as an expansion of the guiding wave function over the velocity space. It is shown that the quantum evolution is preserved in its semiclassical scheme through this change. The case of variable-frequency harmonic oscillator is presented as an example.

Generalized Langevin equation with tempered memory kernel

Abstract We study a generalized Langevin equation for a free particle in presence of a truncated power-law and Mittag-Leffler memory kernel. It is shown that in presence of truncation, the particle from subdiffusive behavior in the short time limit, turns to normal diffusion in the long time limit. The case of harmonic oscillator is considered as well, and the relaxation functions and the normalized displacement correlation function are represented in an exact form. By considering external time-dependent periodic force we obtain resonant behavior even in case of a free particle due to the influence of the environment on the particle movement. Additionally, the double-peak phenomenon in the imaginary part of the complex susceptibility is observed. It is obtained that the truncation parameter has a huge influence on the behavior of these quantities, and it is shown how the truncation parameter changes the critical frequencies. The normalized displacement correlation function for a fractional generalized Langevin equation is investigated as well. All the results are exact and given in terms of the three parameter Mittag-Leffler function and the Prabhakar generalized integral operator, which in the kernel contains a three parameter Mittag-Leffler function. Such kind of truncated Langevin equation motion can be of high relevance for the description of lateral diffusion of lipids and proteins in cell membranes.