Quantum Mechanical Systems Research Articles

Pedagogical moves related to analogy that support a unified understanding of eigentheory concepts in a quantum mechanics class

It is beneficial for quantum mechanics students to have a unified understanding of eigentheory concepts, so they can recognize the shared structure of mathematized phenomena from the different quantum mechanical systems of spin, energy, or position and recognize those as instantiations of the same overarching concept. Quantum mechanics instructors should, therefore, provide opportunities for their class community to develop a shared unified understanding of eigentheory concepts. One such opportunity can arise by engaging students in analogizing eigentheory concepts in one context with those from another context. We investigate the pedagogical moves related to analogies that can be used by a quantum mechanics course instructor to support a class community in developing a shared unified understanding of eigenequations. We analyze classroom data to characterize an instructor’s pedagogical moves as he engaged students in analogical reasoning. Some moves include posing tasks conducive to analogizing; preparing, soliciting, and scaffolding students’ participation in analogical reasoning; using deictic gestures and inscriptions; juxtaposing symbols representing the analogized concepts; and explicitly highlighting the sameness of the analogized concepts. We exemplify these pedagogical moves with analytical descriptions of illustrative class episodes. We discuss how these pedagogical moves can support the class community’s expansion of their common ground by fostering the development of the class’s shared unified understanding of eigentheory concepts. Published by the American Physical Society 2024

Linearly Implicit Conservative Schemes for the Nonlocal Schrödinger Equation

This paper introduces two high-accuracy linearly implicit conservative schemes for solving the nonlocal Schrödinger equation, employing the extrapolation technique. These schemes are based on the generalized scalar auxiliary variable approach and the symplectic Runge–Kutta method. By integrating these advanced methods, the proposed schemes aim to significantly enhance computational accuracy and efficiency, while maintaining the essential conservative properties necessary for accurate physical modeling. This offers a structured approach to handle auxiliary variables, ensuring stability and conservation, while the symplectic Runge–Kutta method provides a robust framework with high accuracy. Together, these techniques offer a powerful and reliable approach for researchers dealing with complex quantum mechanical systems described by the nonlocal Schrödinger equation, ensuring both accuracy and stability in their numerical simulations.

Out-of-time-ordered-correlators for the pure inverted quartic oscillator: classical chaos meets quantum stability

Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of –2⟨x2⟩T ⟨p2⟩T at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.

Quantified asymptotic analysis for the relativistic quantum mechanical system with electromagnetic fields

We study the asymptotic analysis of a relativistic quantum mechanical system interacting with self-consistent electromagnetic fields, with a specific focus on the Maxwell–Klein–Gordon (MKG) system. Firstly, we show that the MKG system is globally well-posed under the Coulomb gauge condition, even in the presence of self-interacting potential. Furthermore, to derive the asymptotic analysis, we consider the non-relativistic and semi-classical regime simultaneously, introducing a single scaling parameter that serves to parameterize both the speed of light and the Planck constant. As the scaling parameter approaches zero, we obtain rigorous and quantified estimates on the asymptotic convergence of the electrostatic MKG system toward the classical Euler–Poisson system. Our analytical framework relies on the modulated energy estimate.

Spinning Systems in Quantum Mechanics: An Overview and New Trends

Spinning Systems in Quantum Mechanics: An Overview and New Trends

Harmonic oscillator in topologically charged deformed gravity space–time and Wu–Yang magnetic monopole

We investigate the quantum dynamics of a harmonic oscillator (HO) system within the framework of topologically charged modified Eddington-inspired Born–Infeld (EiBI) gravity space–time. Additionally, we incorporate the Wu–Yang magnetic monopole (WYMM) into the quantum system and analyze the influences of modified gravity, the topological charge, and WYMM on this HO quantum mechanical system. Using analytical methods, we derive the energy eigenvalues and the corresponding eigenfunctions of the HO. Moreover, we consider a scenario where the modified EiBI-gravity parameter is absent and introduce an inverse square potential, including the WYMM, into the HO system. Our findings show significant deviations in the behavior of the HO system compared to the traditional quantum mechanical HO model, including alterations in the bound-state spectra and eigenfunctions. These results provide valuable insights into the interplay between quantum mechanical problems and alternative gravitational theories.

Neural network representation of quantum systems

Abstract It has been proposed that random wide neural networks near Gaussian process are quantum field theories around Gaussian fixed points. In this paper, we provide a novel map with which a wide class of quantum mechanical systems can be cast into the form of a neural network with a statistical summation over network parameters. Our simple idea is to use the universal approximation theorem of neural networks to generate arbitrary paths in the Feynman’s path integral. The map can be applied to interacting quantum systems / field theories, even away from the Gaussian limit. Our findings bring machine learning closer to the quantum world.

Out-of-time-order correlator as a detector of baryonic phase structure in holographic QCD with instanton

We study the out-of-time-order correlators (OTOC) of Skyrmion as baryon in the D0-D4/D8 model which is expected to be holographically dual to QCD with instantons as D0-branes or with a non-zero theta angle. Baryon states are identified to the excitations of the Skyrmion which are described by a holographic quantum mechanical system in this model. By employing the definition of OTOC in quantum mechanics, we derive the formulas and demonstrate explicitly the numerical calculations of the OTOC. Our calculation illustrates the quantum OTOC with imaginary Lyapunov coefficient indicates the possibly metastable baryonic status in the presence of the instanton while the classical OTOC can not, thus it reveals the instantonic or theta-dependent features of QCD are dominated basically by its quantum properties. Furthermore, the OTOC also implies the baryonic phase becomes really chaotic with real Lyapunov exponent if the instanton charge increases sufficiently which agrees with the unstable baryon spectrum presented in this model. In this sense, we believe the OTOC may be treated as a tool to detect the baryonic phase structure of QCD.

Estimation of equilibration time scales from nested fraction approximations.

We consider an autocorrelation function of a quantum mechanical system through the lens of the so-called recursive method, by iteratively evaluating Lanczos coefficients or solving a system of coupled differential equationsin the Mori formalism. We first show that both methods are mathematically equivalent, each offering certain practical advantages. We then propose an approximation scheme to evaluate the autocorrelation function and use it to estimate the equilibration time τ for the observable in question. With only a handful of Lanczos coefficients as the input, this scheme yields an accurate order of magnitude estimate of τ, matching state-of-the-art numerical approaches. We develop a simple numerical scheme to estimate the precision of our method. We test our approach using several numerical examples exhibiting different relaxation dynamics. Our findings provide a practical way to quantify the equilibration time of isolated quantum systems, a question which is both crucial and notoriously difficult.

A Perspective on the Investigation of Spectroscopy and Kinetics of Complex Molecular Systems with Semiclassical Approaches.

In this Perspective we show that semiclassical methods provide a rigorous hierarchical way to study the vibrational spectroscopy and kinetics of complex molecular systems. The time averaged approach to spectroscopy and the semiclassical transition state theory for kinetics, which have been first adopted and then further developed in our group, provide accurate quantum results on rigorous physical grounds and can be applied even when dealing with a large number of degrees of freedom. In spectroscopy, the multiple coherent, divide-and-conquer, and adiabatically switched semiclassical approaches have practically permitted overcoming issues related to the convergence of results. In this Perspective we demonstrate the possibility of studying the semiclassical vibrational spectroscopy of a molecule adsorbed on an anatase (101) surface, a system made of 51 atoms. In kinetics, the semiclassical transition state theory is able to account for anharmonicity and the coupling between the reactive and bound modes. Our group has developed this technique for practical applications involving the study of phenomena like kinetic isotope effect, heavy atom tunneling, and elusive conformer lifetimes. Here, we show that our multidimensional anharmonic quantum approach is able to tackle on-the-fly the thermal kinetic rate constant of a 135 degree-of-freedom system. Overall, semiclassical methods open up the possibility to describe at the quantum mechanical level systems characterized by hundreds of degrees of freedom leading to the accurate spectroscopic and kinetic description of biomolecules and complex molecular systems.

A 3D field-theoretic example for Hodge theory

We focus on the continuous symmetry transformations for the three (2 + 1)-dimensional (3D) system of a combination of the free Abelian 1-form and 2-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We establish that this combined system is a tractable field-theoretic model of Hodge theory. The symmetry operators of our present system provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. Our present investigation is important in the sense that, for the first time, we are able to establish an odd dimensional (i.e., D = 3) field-theoretic system to be an example for Hodge theory (besides earlier works on a few interesting (0 + 1)-dimensional (1D) toy models as well as a set of well-known SUSY quantum mechanical systems of physical interest). For the sake of brevity, we have purposely not taken into account the 3D Chern-Simon term for the Abelian 1-form gauge field in our theory which allows the mass as well as the gauge invariance to co-exist together.

Late time dynamics in SUSY saddle-dominated scrambling through higher-point OTOC

In this article, we propose higher-point out-of-time-order correlators (OTOCs) as a tool to differentiate chaotic from saddle-dominated dynamics in late times. As a model, we study the scrambling dynamics in supersymmetric quantum mechanical systems. Using the eigenstate representation, we define the 2N-point OTOC using two formalisms, namely the ’Tensor Product formalism’ and the ’Partner Hamiltonian formalism’. We analytically find that the 2N-point OTOC for the supersymmetric 1D harmonic oscillator is in exact agreement with that of the 1D bosonic harmonic oscillator system. We show that the higher-point OTOC is a more sensitive measure of scrambling than the usual 4-point OTOC. To demonstrate this, we analyze a supersymmetric sextic 1D oscillator, for which the bosonic partner system has an unstable saddle in the phase space, while the saddle is absent in the fermionic counterpart. For such a system, we show that the saddle-dominated scrambling, higher anharmonic potential effects, and the supersymmetric OTOC exhibit similar dynamics due to supersymmetry constraints. Finally, we illustrate that the late-time dynamics of the higher-point OTOC become oscillatory after the peak for saddle-dominated scrambling and anharmonic oscillator systems. We propose the higher-point OTOC as a probe of late-time dynamics in non-chaotic systems that exhibit fast early-time scrambling.

Aharonov-Bohm experiments in magnetic field without the Landau levels

We discuss the two-slit experiment and the Aharonov-Bohm (AB) experiment in the magnetic field, where an electron produces so called synchrotron radiation. In other words, the photons are emitted from the points of the electron trajectory and it means that the trajectory of electron is visible in the synchrotron radiation spectrum. The axiomatic system of quantum mechanics does not enable to define the trajectory of the elementary particle. The two-slit experiment and the AB experiment in magnetic field are the missing experiments of quantum mechanics. The extension of the discussion to the cosmical rays moving in the magnetic field of the Saturn magnetosphere and its rings is mentioned. It is related to the probe CASSINI.

QM/CG-MM: Systematic Embedding of Quantum Mechanical Systems in a Coarse-Grained Environment with Accurate Electrostatics.

Quantum Mechanics/Molecular Mechanics (QM/MM) can describe chemical reactions in molecular dynamics (MD) simulations at a much lower cost than ab initio MD. Still, it is prohibitively expensive for many systems of interest because such systems usually require long simulations for sufficient statistical sampling. Additional MM degrees of freedom are often slow and numerous but secondary in interest. Coarse-graining (CG) is well-known to be able to speed up sampling through both reduction in simulation cost and the ability to accelerate the dynamics. Therefore, embedding a QM system in a CG environment can be a promising way of expediting sampling without compromising the information about the QM subsystem. Sinitskiy and Voth first proposed the theory of Quantum Mechanics/Coarse-grained Molecular Mechanics (QM/CG-MM) with a bottom-up CG mapping. Mironenko and Voth subsequently introduced the DFT-QM/CG-MM formalism to couple a Density Functional Theory (DFT) treated QM system and to an apolar environment. Here, we present a more complete theory that addresses MM environments with significant polarity by explicitly accounting for the electrostatic coupling. We demonstrate our QM/CG-MM method with a chloride-methyl chloride SN2 reaction system in acetone, which is sensitive to solvent polarity. The method accurately recapitulates the potential of mean force for the substitution reaction, and the reaction barrier from the best model agrees with the atomistic simulations within sampling error. These models also have generalizability. In two other reactive systems that they have not been trained on, the QM/CG-MM model still achieves the same level of agreement with the atomistic QM/MM models. Finally, we show that in these examples the speed-up in the sampling is proportional to the acceleration of the rotational dynamics of the solvent in the CG system.

Gauged permutation invariant matrix quantum mechanics: partition functions

The Hilbert spaces of matrix quantum mechanical systems with N × N matrix degrees of freedom X have been analysed recently in terms of SN symmetric group elements U acting as X → UXUT. Solvable models have been constructed uncovering partition algebras as hidden symmetries of these systems. The solvable models include an 11-dimensional space of matrix harmonic oscillators, the simplest of which is the standard matrix harmonic oscillator with U(N) symmetry. The permutation symmetry is realised as gauge symmetry in a path integral formulation in a companion paper. With the simplest matrix oscillator Hamiltonian subject to gauge permutation symmetry, we use the known result for the micro-canonical partition function to derive the canonical partition function. It is expressed as a sum over partitions of N of products of factors which depend on elementary number-theoretic properties of the partitions, notably the least common multiples and greatest common divisors of pairs of parts appearing in the partition. This formula is recovered using the Molien-Weyl formula, which we review for convenience. The Molien-Weyl formula is then used to generalise the formula for the canonical partition function to the 11-parameter permutation invariant matrix harmonic oscillator.

Convergence of resonant state expansions for transmission in double delta resonators

We investigate the transmission features of symmetric double Dirac delta resonators using a non-Hermitian quantum approach, based on resonant state expansions involving complex energy eigenvalues. We focus on the convergence properties of these expansions, which involve a sum of resonance and anti-resonance terms We demonstrate that systems with transmission profiles featuring sharp and isolated resonances converge more rapidly, whereas those with overlapping resonances exhibit slower convergence. We also show that for the former systems, by taking into account only the resonance terms, we can accurately describe the transmission coefficient as a sum of Breit-Wigner resonances, each distinctly characterized by its energy and width. We demonstrate that there is a one-to-one correlation between transmission peaks and the resonance energies of the system. We also compare the convergence properties of transmission using Mittag-Leffler expansions, noting their slower rates compared to resonant state expansions. These findings emphasize the advantage of using non-Hermitian resonant expansions for analyzing quantum mechanical systems, providing a clearer understanding of systems characterized by resonant features.

Noise intensity of a Markov chain.

Stochastic transitions between discrete microscopic states play an important role in many physical and biological systems. Often these transitions lead to fluctuations on a macroscopic scale. A classic example from neuroscience is the stochastic opening and closing of ion channels and the resulting fluctuations in membrane current. When the microscopic transitions are fast, the macroscopic fluctuations are nearly uncorrelated and can be fully characterized by their mean and noise intensity. We show how, for an arbitrary Markov chain, the noise intensity can be determined from an algebraic equation, based on the transition rate matrix; these results are in agreement with earlier results from the theory of zero-frequency noise in quantum mechanical and classical systems. We demonstrate the validity of the theory using an analytically tractable two-state Markovian dichotomous noise, an eight-state model for a calcium channel subunit (De Young-Keizer model), and Markov models of the voltage-gated sodium and potassium channels as they appear in a stochastic version of the Hodgkin-Huxley model.

Time Evolution of Biochemical Materials: Markov chains and MarkovStates Models

The biochemical materials are described in terms of the opportune Hierarchical Markov-State Model and of the originating chain(s). The time evolution of the equations of motion of the Markov chain is controlled; to this aim, the transitions form the unrestrained simulations and those between the local Markov-State Models are compared. The formalisms of quantum-mechanical systems are applied in the opportune measure spaces. The ergodicity of the Markov chains is controlled. The numerical simulations, the properties to be requested on numerical approximations are studied. As a result, the ergodicity of the Mar

Projection hypothesis in the setting for the quantum Jarzynski equality

Projective quantum measurement is a theoretically accepted process in modern quantum mechanics. However, its projection hypothesis is widely regarded as an experimentally established empirical law. In this paper, we combine a previous result regarding the realization of a Hamiltonian process of the projection hypothesis in projective quantum measurement, where the complete set of the orbital observables of the center of mass of a macroscopic quantum mechanical system is restricted to a set of mutually commuting classical observables, and a previous result regarding the work required for an event reading (i.e. the informatical process in projective quantum measurement). Then, a quantum thermodynamic scheme is proposed for experimentally testing these two mutually independent theoretical results of projective quantum measurement simultaneously.

Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 1. Systems with cylindric symmetry

Cylindrically symmetric quantum mechanical systems with position dependent masses admitting at least one second order integral of motion are classified. It is proved that there exist 68 such systems which are inequivalent. Among them there are thirty superintegrable and twelve maximally superintegrable ones. The arbitrary elements of the corresponding Hamiltonians (i.e.,masses and potentials) are presented explicitly.