Stationary Spectra Research Articles

Fluorescence by incoherently pumped polar quantum dot placed in a low-frequency monochromatic field

Abstract We studied spectral properties of high-frequency fluorescent radiation from a two-level quantum system with broken inversion spatial symmetry, which can be implemented as a model of a one-electron two-level asymmetric polar semiconductor quantum dot whose electric dipole moment operator has permanent unequal diagonal matrix elements. The case of the excitation of this system by incoherent pumping of some sort supplemented with an additional driving monochromatic field, which frequency is much lower than the optical transition frequency of the quantum dot, was considered. An analytical expression for the fluorescence spectrum as a function of the amplitude and frequency of the monochromatic driving field, as well as of the pumping intensity, was derived. We also discussed feasible practical applications of the effects under study in the field of quantum optics and optoelectronics for the generation of high-frequency radiation, possessing stationary equidistant spectrum, and for the control of the radiative properties of such nanodevices as spasers (dipole nanolasers).

Structural damage detection based on structural macro-strain mode shapes extracted from non-stationary output responses

Abstract Long-gauge fiber Bragg grating strain sensors have been widely employed because of their broader measuring range and higher sensitivity. However, current structural damage detection methods using macro-strain modal parameters are based on structural frequency response function or stationary power spectrum density, which are not applicable to non-stationary responses. To overcome this limitation, an improved method is proposed in this paper for structural damage detection based on structural macro-strain responses under unknown multi-point non-stationary excitations. First, a new concept of macro-strain energy spectrum transmissibility (MEST) is proposed using structural non-stationary macro-strain responses, and it is derived that MEST at a certain system pole equals the ratio of macro-strain mode shape. Then, the singular value decomposition technique is adopted for the MEST matrix to identify structural natural frequencies and macro-strain mode shapes. Finally, two damage detection indicators are constructed based on the identified normalized macro-strain mode shape (NMMS). The first indicator is the difference in structural NMMS before and after structural damage. The second one is based on the curvatures of structural NMMS, which can be used for structures without intact baseline. Numerical verifications are conducted to identify beam-type structural damage under multi-point non-stationary excitations or vehicle loads. Five damage scenarios with different measurement noise levels are investigated, and damage detection results validate the effectiveness of the proposed method.

Identification of full-field wind loads on buildings using a mechanism-inspired recursive convolutional neural network with partial structural responses

Indirect identification approaches through structural responses have proven effective for wind load estimation in real-world engineering. Currently, methods for identifying wind loads mainly rely on theoretical inverse identification, with rare research based on the mapping relationship between structural responses and wind loads through machine learning. In this paper, a scheme for identifying full-field wind loads using a recursive convolutional neural network (CNN) inspired by physical mechanisms is proposed. The recursive form of the network, as well as the inspiration for its inputs and outputs, is inspired by the spatial correlation and the mapping relationship between wind loads and structural responses. Thus, the network inputs comprise a fusion of structural acceleration and inter-story displacement responses, while the network outputs represent the independent wind loads on structures. Notably, mismatch test is employed by the network, wherein the training and testing datasets originate from entirely different sources. Specifically, during training, Gaussian white noises that simulate wind loads are utilized, while real wind load data are used for testing. The generalization of the proposed scheme is demonstrated through the identification of full-field wind loads generated by different stationary or non-stationary wind spectra of the 76-story wind-excited benchmark building. Furthermore, the proposed scheme is validated by identifying the full-field wind loads of a 67-story shear wall structure with wind tunnel test data.

Vertical time-of-flight spectroscopy of ultracold neutrons

This paper describes the mathematical basis for analysis of vertical time-of-flight (TOF) measurements with ultracold neutrons (UCN). In contrast to conventional horizontal TOF, this enables spectroscopy of UCN incident on the chopper with kinetic energies below the neutron optical potential of the detector entrance window. Much of the treatment presented here is general, and may be applied to horizontal and vertical TOF data of stationary and time-dependent spectra. Transformation rules are given for the conversion of vertical TOF spectra into other spectra of interest, and some aspects of detection efficiency are discussed that are common to all vertical TOF measurements using detectors with entrance windows. Throughout the paper, examples are provided using experimental data from measurements at the SUN-2 prototype superthermal UCN source. The technical details of these experiments and a comprehensive characterization of the source will be published by the present authors in a later paper.

A multiple time scale approach for anisotropic inertial wave turbulence

Wave turbulence is the study of the long-time statistical behaviour of equations describing a set of weakly nonlinear interacting waves. Such a theory, which has a natural asymptotic closure, allows us to probe the nature of turbulence more deeply than the exact Kolmogorov laws by rigorously proving the direction of the cascade and the existence of an inertial range, predicting stationary spectra for conserved quantities, or evaluating the Kolmogorov constant. An emblematic example is given by fast rotating fluids for which a wave turbulence theory has been derived by Galtier (Phys. Rev. E, vol. 68, issue 1, 2003, p. 015301). This work involves non-trivial analytical developments for a problem that is anisotropic by nature. We propose here a new path for the derivation of the kinetic equation by using the anisotropy at the beginning of the analysis. We show that the helicity basis is not necessary to obtain the wave amplitude equation for the canonical variables that involve a combination of poloidal and toroidal fields. The multiple time scale method adapted to this anisotropic problem is then used to derive the kinetic equation that is the same as the original work when anisotropy is eventually taken into account. This result proves the commutativity between asymptotic closure and anisotropy. In addition, the multiple time scale method informs us that the kinetic equation can be derived without imposing restrictions on the probability distribution of the wave amplitude such as quasi-Gaussianity, or on the phase such as random phase approximation that naturally occurs dynamically.

Spectroscopic Characterization of Thiacarbocyanine Dye Molecules Adsorbed on Hexagonal Boron Nitride: a Time-Resolved Study.

Physisorption on hexagonal boron nitride (hBN) gained interest over the years thanks to its properties (chemically and thermally stable, insulating properties, etc.) and similarities to the well-known graphene. A recent study showed flat-on adsorption of several cationic thiacarbocyanine dyes on hBN with a tendency to form weakly coupled H- or I-type aggregates, while a zwitterionic thiacarbocyanine dye rather led to a tilted adsorption. With this in-depth time-resolved study using the TC-SPC technique, we confirm the results proven by adsorption isotherms, atomic force microscopy, and stationary state spectroscopy combined with molecular mechanics simulations and estimation of the corresponding exciton interaction. The absence of a systematic trend for the dependence of the decay times, normalized amplitudes of the decay components, and contribution of different components to the stationary emission spectra upon the emission wavelength observed for all studied dyes and coverages suggests the occurrence of a single emitting species. At low coverage levels, the non-mono-exponential character of the decays was attributed to adsorption on different sites characterized by different intramolecular rotational freedom or energy transfer to nonfluorescent traps or a combination of both. The difference between the decay rates of the four dyes reflects a different density of the nonfluorescent traps. Although the decay time of the unquenched dyes was in the order of magnitude of that of dye monomers in a rigid environment, it is also compatible with weakly coupled aggregates such as proposed earlier based on the stationary spectra. Hence, the adsorption leads to a rigid environment of the dyes, blocking internal conversion. Increasing the concentration of the dye solution from which the adsorption on hBN occurs increases not only the coverage of the hBN surface but also the extent of energy transfer to nonfluorescent traps. For TDC (5,5-dichloro-3-3'-diethyl-9-ethyl-thiacarbocyanine) and TD2 (3-3'-diethyl-9-ethyl-thiacarbocyanine), besides direct energy transfer to traps, exciton hopping between dye dimers followed by energy transfer to these traps occurs, which resulted in a decreasing decay time of the longest decaying component. For all dyes, it was also possible to analyze the fluorescence decays as a stretched exponential as would be expected for energy transfer to randomly distributed traps in a two-dimensional (2D) geometry. This analysis yielded a fluorescence decay time of the unquenched dyes similar to the longest decay time obtained by analysis of the fluorescence decays as a sum of three of four exponentials.

Excited State Dynamics of Alizarin Red S Nanoparticles in Solution.

Alizarin red S is a sulfonated, water-soluble derivative of alizarin. This work presents femtosecond studies of alizarin red S (ARS) nanoparticles in comparison to ARS in aqueous solution and to alizarin in DMSO. The femtosecond studies cover a probing spectral range of 350-750 nm using different excitation wavelengths, taking into account the variation of the absorption spectra with the pH values of the solvent. Stationary absorption spectra show slight differences between solution and nanoparticles. Excitation at 530 nm results in low and noisy responses, therefore, we additionally recorded transient spectra of the nanoparticles at λex = 267 nm. While the results in DMSO are comparable to previous studies in non-aqueous solvents, we report a relatively fast relaxation of 14 ps in [La(OH)2][ARS] nanoparticles in aqueous solution after excitation at 530 nm, which is similar to Na(ARS) solution (19 ps). The dynamics changed with lower pH, but still without significant differences between nanoparticles and solution. We propose [La(OH)2][ARS] nanoparticles as a suitable alternative to dissolved molecules with similar spectroscopic properties, for example, with regard to biomarker applications.

An effective semilocal model for wave turbulence in two-dimensional nonlinear optics

The statistical evolution of ensembles of random, weakly interacting waves is governed by wave kinetic equations (WKEs). To simplify the analysis, one frequently works with reduced differential models of the wave kinetics. However, the conditions for deriving such reduced models are seldom justified self-consistently. Here, we derive a reduced model for the wave kinetics of the Schrödinger–Helmholtz equations in two spatial dimensions, which constitute a model for the dynamics of light in a spatially nonlocal, nonlinear optical medium. This model has the property of sharply localizing the frequencies of the interacting waves into two pairs, allowing for a rigorous and self-consistent derivation of what we term the semilocal approximation model (SLAM) of the WKE. Using the SLAM, we study the stationary spectra of Schrödinger–Helmholtz wave turbulence, and characterize the spectra that carry energy downscale, and waveaction upscale, in a forced-dissipated setup. The latter involves a nonlocal transfer of waveaction, in which waves at the forcing scale mediate the interactions of waves at every larger scale. This is in contrast to the energy cascade, which involves local scale-by-scale interactions, familiar from other wave turbulent systems and from classical hydrodynamical turbulence.

Development of High-Temperature Superconducting Coil for Solid-State NMR Experiments

Abstract A high-temperature superconducting (HTS) coil, in which YBa2Cu3O7-δ is vapor-deposited on the surface of a sapphire plane, has been developed to increase signal-to-noise ratios (SNR) in solid-state nuclear magnetic resonance (NMR) measurements. The Q-factor of the NMR probe is found to be 16,125 at the resonant frequency of 38.525 MHz. A single sharp signal, full width at half maximum of 20 ppm, can be detected by the HTS coil in a stationary solid-state 79Br NMR spectrum of a polycrystalline potassium bromide at the temperature of 9 K. Compared with the results for an isomorphic copper coil, an improvement of a factor of 6.6 is obtained for SNR, indicating that accumulation time of NMR experiments can be reduced to 1/44.

Breaking of Internal Waves Parametrically Excited by Ageostrophic Anticyclonic Instability

Abstract A gradient-wind balanced flow with an elliptic streamline parametrically excites internal inertia-gravity waves through ageostrophic anticyclonic instability (AAI). This study numerically investigates the breaking of internal waves and the following turbulence generation resulting from the AAI. In our simulation, we periodically distort the calculation domain following the streamlines of an elliptic vortex and integrate the equations of motion using a Fourier spectral method. This technique enables us to exclude the overall structure of the large-scale vortex from the computation and concentrate on resolving the small-scale waves and turbulence. From a series of experiments, we identify two different scenarios of wave breaking conditioned on the magnitude of the instability growth rate scaled by the buoyancy frequency λ/N. First, when , the primary wave amplitude excited by AAI quickly goes far beyond the overturning threshold and directly breaks. The resulting state is thus strongly nonlinear turbulence. Second, if , weak wave–wave interactions begin to redistribute energy across frequency space before the primary wave reaches a breaking limit. Then, after a sufficiently long time, the system approaches a Garrett–Munk-like stationary spectrum, in which wave breaking occurs at finer vertical scales. Throughout the experimental conditions, the growth and decay time scales of the primary wave energy are well correlated. However, since the primary wave amplitude reaches a prescribed limit in one scenario but not in the other, the energy dissipation rates exhibit two types of scaling properties. This scaling classification has similarities and differences with D’Asaro and Lien’s wave–turbulence transition model. Significance Statement Due to the gradients in buoyancy and pressure, density-stratified seawater supports oscillatory vertical motion called internal waves. When waves significantly skew a density isosurface, dense water lifts over lighter water resulting in gravitational instability and high energy dissipation. In this wave-breaking process, seawater is vertically mixed, transporting heat and nutrients essential to maintain Earth’s climate and ecosystems. This study investigates the generation and breaking of ocean internal waves in a novel numerical simulation setup; we temporally distort the model shape to emulate the wave excitation forced by a larger-size horizontal eddy, a ubiquitous situation at O(1–10) km scales in the upper ocean. The simulation results exhibit two unique wave-breaking scenarios with distinct scaling features in turbulence energy dissipation rates.

Inertia-gravity-wave diffusion by geostrophic turbulence: the impact of flow time dependence

The scattering of three-dimensional inertia-gravity waves by a turbulent geostrophic flow leads to the redistribution of their action through what is approximately a diffusion process in wavevector space. The corresponding diffusivity tensor was obtained by Kafiabad et al. (J. Fluid Mech., vol. 869, 2019, R7) under the assumption of a time-independent geostrophic flow. We relax this assumption to examine how the weak diffusion of wave action across constant-frequency cones that results from the slow time dependence of the geostrophic flow affects the distribution of wave energy. We find that the stationary wave-energy spectrum that arises from a single-frequency wave forcing is localised within a thin boundary layer around the constant-frequency cone, with a thickness controlled by the acceleration spectrum of the geostrophic flow. We obtain an explicit analytic formula for the wave-energy spectrum which shows good agreement with the results of a high-resolution simulation of the Boussinesq equations.

Particle acceleration at ultrarelativistic, perpendicular shock fronts

ABSTRACT Using an eigenfunction expansion to solve the transport equation, complemented by Monte Carlo simulations, we show that ultrarelativistic shocks can be effective particle accelerators even when they fail to produce large amplitude turbulence in the downstream plasma. This finding contradicts the widely held belief that a uniform downstream magnetic field perpendicular to the shock normal inhibits acceleration by the first-order Fermi process. In the ultrarelativistic limit, we find a stationary power-law particle spectrum of index s = 4.17 for these shocks, close to that predicted for a strictly parallel shock.

Non-Markovian transient spectroscopy in cavity QED

We theoretically analyze measurements of the transient field leaving a cavity as a tool for studying non-Markovian dynamics in cavity quantum electrodynamics (QED). Combined with a dynamical decoupling pulse sequence, transient spectroscopy can be used to recover spectral features that may be obscured in the stationary cavity transmission spectrum due to inhomogeneous broadening. The formalism introduced here can be leveraged to perform in situ noise spectroscopy, revealing a robust signature of quantum noise arising from non-commuting observables, a purely quantum effect.

Low-Lying Excited States of Natural Carotenoids Viewed by Ab Initio Methods.

Low-lying excited states of carotenoids (the optically dark 2Ag- and bright 1Bu+) are deeply involved in energy transfer processes in photosynthetic antennas, such as light harvesting and non-photochemical quenching. Though any ab initio modeling of these phenomena has to rely on precise energies of the carotenoid electronic states, the accurate evaluation of these states remains a challenging problem due to their different natures. The paper aims to study the accuracy of the excitation energies of the low-lying excited states of certain open- and closed-chain carotenoids obtained by a state-of-the-art multireference approach for electronic structure calculation. Here, density matrix renormalization group SCF (DMRGSCF) and a perturbative approach based on driven similarity renormalization group second-order multireference perturbation theory (DSRG-MRPT2) were used to treat the static and dynamic correlation, respectively. Nuclear geometries of the electronic states were optimized with DFT-based approaches. It is demonstrated that spin-flip TD-DFT can replace multiconfigurational methods for the geometry optimization of the 2Ag- state but not for the calculation of the excitation energy. Adiabatic excitation energies to the 1Bu+ state were shown to be within a margin of 1000 cm-1 with an appropriate flow parameter value. Adiabatic excitation energies to the 2Ag- state for the open-chain carotenoids lie within a range of experimental values (taking into account the broad range of experimental estimates); for the closed-chain ones, the error does not exceed 2000 cm-1. Ab initio stationary (1Ag- → 1Bu+) and transient (2Ag- → 1Bu+) absorption spectra were modeled for violaxanthin and lycopene, and these spectra showed good agreement with the experimental ones both in terms of the vibronic structure and the transition energies.

Absorption and stationary fluorescent spectra of molecular sensors in solution: A computational study

In this study we report the application and the extension of a theoretical-computational method for the modelling of absorption and stationary emission (electronic) spectra of flexible and easily polarizable chromophores embedded in complex environments. The method, similar but computationally less expensive than the most popular Quantum Mechanics/Molecular Mechanics (QM/MM) approaches, is based on a combination of Molecular Dynamics simulations, Quantum-Chemical calculations and elementary statistical mechanics. As a test case we have addressed, in the present study, the spectral features of two solvated chromophores, experimentally shown by our group to be very sensitive to the solvent (environment) polarity, formed by a 4-[4-(1-dimethylamino)phenyl]-pyridine moiety linked to an uracil group through an alkyl spacer of different lengths. The method, despite the involved approximations and intrinsic drawbacks, is able of reproducing both the absorption and emission spectra of the above system as well as the sharp dependency of the spectral observables on the environment polarity. In this respect it might represent an efficient tool, alternative to more accurate QM/MM-based approaches, in cases where their application could be made computationally very expensive.

Опыт реконструкции облика человека раннего неолита из погребения могильника Шумилиха (Южное Приангарье)

This paper presents the results of a reconstruction of the appearance of a hunter-gatherer of the Kitoi culture from a burial at the Shumilikha cemetery in the Southern Angara region. The excavations were carried out in 1999. In the grave, the walls of which were coated with clay, the remains of a human 25–35 years old were found. Under the skeleton, which was sprinkled with “ocher” (grated hematite rock), traces of a wooden construction were recorded. The grave goods (more than a hundred items) included such interesting finds as the kernels of fishhooks, a bone spoon, beads made of bone paste, as well as two items made of white jade: a ring and a cylindrical bead. There is a reliable radiocarbon date for the human bone, which, taking into account the correction for the reservoir effect, shows an age of 7,003±106 cal BP and allows you to determine the age of this complex by the Early Neolithic. Modern software for 3D modeling and visualization opens up great opportunities, so a virtual method was chosen to reconstruct the appearance of a human from a burial. First, all the bones of the skeleton were scanned, for which two types of 3D equipment were used: a stationary scanner RangeVision Spectrum based on structured light photogrammetry (for the skull and small bones) and a hybrid handheld scanner with structured LED and infrared illumination Shining 3D Einscan H (for large bones). The scan results are processed in RangeVision's ScanCenter NG using built-in algorithms. Filling voids, smoothing surfaces, simplifying 3D models, and correcting mesh defects are done manually at the stage of model postprocessing. For further work, the resulting models were transferred to the stl format. Facial reconstruction was carried out in Blender and ZBrush in several stages: 1) importing and combining models; 2) modeling of lost teeth and reconstruction of the occlusion; 3) CAD modeling of a set of soft tissue depth markers and their positioning by craniometric points; 4) modeling of auxiliary circuits for nose reconstruction and eyeball setting; 5) modeling of masticatory muscles; 6) modeling of facial muscles and fat sacs to simplify the production of a basic head model; 7) modeling the base mesh of the head; 8) digital sculpting of the head; 9) detailing of the digital sculpture; 10) rendering of the model and preparing the model for printing. As a result, a digital model of the appearance of an Early Neolithic hunter-gatherer from a burial at the Shumilikha cemetery was obtained.

Опыт реконструкции облика человека раннего неолита из погребения могильника Шумилиха (Южное Приангарье)

This paper presents the results of a reconstruction of the appearance of a hunter-gatherer of the Kitoi culture from a burial at the Shumilikha cemetery in the Southern Angara region. The excavations were carried out in 1999. In the grave, the walls of which were coated with clay, the remains of a human 25–35 years old were found. Under the skeleton, which was sprinkled with “ocher” (grated hematite rock), traces of a wooden construction were recorded. The grave goods (more than a hundred items) included such interesting finds as the kernels of fishhooks, a bone spoon, beads made of bone paste, as well as two items made of white jade: a ring and a cylindrical bead. There is a reliable radiocarbon date for the human bone, which, taking into account the correction for the reservoir effect, shows an age of 7,003±106 cal BP and allows you to determine the age of this complex by the Early Neolithic. Modern software for 3D modeling and visualization opens up great opportunities, so a virtual method was chosen to reconstruct the appearance of a human from a burial. First, all the bones of the skeleton were scanned, for which two types of 3D equipment were used: a stationary scanner RangeVision Spectrum based on structured light photogrammetry (for the skull and small bones) and a hybrid handheld scanner with structured LED and infrared illumination Shining 3D Einscan H (for large bones). The scan results are processed in RangeVision's ScanCenter NG using built-in algorithms. Filling voids, smoothing surfaces, simplifying 3D models, and correcting mesh defects are done manually at the stage of model postprocessing. For further work, the resulting models were transferred to the stl format. Facial reconstruction was carried out in Blender and ZBrush in several stages: 1) importing and combining models; 2) modeling of lost teeth and reconstruction of the occlusion; 3) CAD modeling of a set of soft tissue depth markers and their positioning by craniometric points; 4) modeling of auxiliary circuits for nose reconstruction and eyeball setting; 5) modeling of masticatory muscles; 6) modeling of facial muscles and fat sacs to simplify the production of a basic head model; 7) modeling the base mesh of the head; 8) digital sculpting of the head; 9) detailing of the digital sculpture; 10) rendering of the model and preparing the model for printing. As a result, a digital model of the appearance of an Early Neolithic hunter-gatherer from a burial at the Shumilikha cemetery was obtained.

A new method to specify the mitigated acoustic test conditions from flight telemetry based on the equivalence of vibro-acoustic response

Ground acoustic tests using stationary sound pressure level spectrum have been conducted to verify the spacecraft survivability against acoustic environment acting on a spacecraft during launch, which is a non-stationary and random dynamic load. In general, a stationary spectrum used in ground acoustic test is traditionally determined by a method called maximax spectrum, which is the enveloped spectrum of time varying non-stationary short-time Fourier transform. However, the maximax spectrum is more or less an excessively conservative test condition because this spectrum focuses on processing of a time-varying acoustic signal itself to extract maximum value, rather than on how the vibro-acoustic response of an excited structure is. In this paper, a new method is proposed to specify a stationary spectrum equivalent to a structural vibro-acoustic response under a non-stationary and random acoustic environment based on extreme response spectrum and fatigue damage spectrum. This proposed method was applied to flight telemetry of both liquid- and solid-propellant launch vehicles developed by JAXA, to show its effect to mitigate the acoustic test conditions compared to the maximax spectrum while maintaining the equivalence of the structural vibro-acoustic response. Furthermore, the maximum predicted environment, which is the statistical upper percentiles of the flight telemetry of eight liquid-propellant launch vehicles, by the proposed method achieved a mitigation of about 2.5 and 6.8 dB in the extreme stress and cumulative fatigue, respectively, compared to the that which is calculated by the conventional maximax spectrum.

Principles of modeling the fluorescence spectral dynamics of dye molecules in solutions

FSDS is Fluorescence Spectral Dynamics Simulator, designed to investigate photoinduced charge transfer and its manifestations in time-resolved fluorescence spectra of a dye in a solvent. The project is implemented in C, using the MPI library. The computer model simulates stationary absorption and fluorescence spectra, as well as the time evolution of population distributions along the solvent reaction coordinate, to calculate spectral dynamics. Excitation and relaxation of intramolecular high-frequency vibrations are described at the quantum level. FSDS can fit the physical parameters of the dyes to experimental data. The dye excitation is consistently described accounting for the finite duration of the pump pulse. The program code implements two evolutionary models: a spin-boson model within the harmonic potential and a stochastic model based on the Smoluchowski diffusion operator for an arbitrary free energy surface. Computer simulation shows the numerical coincidence of the calculations performed in the framework of these models for the harmonic free energy surface. The detail description of the code and numerical schemes are presented. Program summaryProgram Title: Fluorescence Spectral Dynamics Simulator (FSDS)CPC Library link to program files:https://doi.org/10.17632/d39rrrkvhr.1Code Ocean capsule:https://codeocean.com/capsule/6544976Licensing provisions: GPLv3Programming language: CSupplementary material: Sample configuration files, User ManualNature of problem: The time-resolved spectra contain reach information about photochemical processes. However, the chemically important information about separate elementary processes is entangled due to they proceed in parallel. Time-resolved spectra modeling allows extracting information about these processes. The presented program code makes it possible to determine the relaxation characteristics of the solvent and the times of intramolecular relaxation.Solution method: Finite-difference time-domain and Brownian dynamics simulation methods.

Effective mass of the discrete values as a hidden feature of the one-dimensional harmonic oscillator model: Exact solution of the Schrödinger equation with a mass varying by position

In this paper, exactly solvable model of the quantum harmonic oscillator is proposed. Wave functions of the stationary states and energy spectrum of the model are obtained through the solution of the corresponding Schrödinger equation with the assumption that the mass of the quantum oscillator system varies with position. We have shown that the solution of the Schrödinger equation in terms of the wave functions of the stationary states is expressed by the pseudo Jacobi polynomials and the mass varying with position depends from the positive integer [Formula: see text]. As a consequence of the positive integer [Formula: see text], energy spectrum is not only non-equidistant, but also there are only a finite number of energy levels. Under the limit, when [Formula: see text], the dependence of effective mass from the position disappears and the system recovers known non-relativistic quantum harmonic oscillator in the canonical approach where wave functions are expressed by the Hermite polynomials.