Exact Energy Spectrum Research Articles

Quantum dot in a graphene layer with topological defects

In this paper we use an exactly solvable model in order to investigate a magnetization and persistent current of massless Dirac fermions confined in a quantum dot in a graphene layer with a topological defect. We study the low-energy electronic spectrum of a graphene layer structure using the continuum approach and introduce a disclination by means of the geometric theory of defects. The model of the confining potential is introduced into the system via the Dirac oscillator, which provides a harmonic confinement for quasiparticles, in order to study a behaviour of quantum dots in graphene. The exact energy spectrum and wave functions are obtained analytically for the model, in the case where a thin magnetic flux confined to the dot centre, and the arising of persistent currents is investigated. A uniform magnetic field is introduced in a quantum dot, and this model is used to study persistent currents and magnetization in the graphene dot. We have derived exact expressions both for the magnetic moment and for the current carried by quasiparticle states.

Anisotropic Rabi model

We define the anisotropic Rabi model as the generalization of the spin-boson Rabi model: The Hamiltonian system breaks the parity symmetry; the rotating and counter-rotating interactions are governed by two different coupling constants; a further parameter introduces a phase factor in the counter-rotating terms. The exact energy spectrum and eigenstates of the generalized model is worked out. The solution is obtained as an elaboration of a recent proposed method for the isotropic limit of the model. In this way, we provide a long sought solution of a cascade of models with immediate relevance in different physical fields, including i) quantum optics: two-level atom in single mode cross electric and magnetic fields; ii) solid state physics: electrons in semiconductors with Rashba and Dresselhaus spin-orbit coupling; iii) mesoscopic physics: Josephson junctions flux-qubit quantum circuits.

An analytical approach for the energy spectrum and optical properties of gated bilayer graphene

An analytical approach is developed to access the exact energy spectrum, wave functions, dipole matrix element (Mfi) and absorption spectra (A(ω)) of gated Bernal bilayer graphene.

Relativistic oscillator model with spin for nucleon resonances

Abstract The relativistic three-body problem is approached via the extension of the SL(2, C) group to the Sp(4, C) one. In terms of Sp(4, C) spinors, a Dirac-like equation with three-body kinematics is composed. After introducing the linear in coordinates interaction, it describes the spin-1/2 oscillator. For this system, the exact energy spectrum is derived and then applied to fit the Regge trajectories of baryon N-resonances in the (E 2, J) plane. The model predicts linear trajectories at high total energy E with some form of nonlinearity at low E.

Exact Spin Coherent State Path Integral for a Damped Two-Level Atom in an Electromagnetic Wave

The spin coherent state path integral describing the dynamics of a two-level system interacting with electromagnetic wave of circular polarization is considered. The propagator is first written in the standard form by replacing the spin by a unit vector aligned along the polar and azimuthal directions. Then it is determined exactly using a simple transformations. Thus, the exact energy spectra with corresponding wave functions are deduced.

Exact Landau spectrum and wave functions of biased AA-stacked multilayer graphene

A novel model is presented to obtain exact energy spectrum, Landau levels (LLs) and wave functions in AA-stacking N-layer Graphene under the influence of perpendicular electric and magnetic fields. A particular unitary transformation to deal with the broken mirror symmetry caused by a perpendicular electric field can transform an AA-stacking N-layer graphene into N independent graphene-like layers. The exact energy spectrum of a graphene-like layer is equal to E=EMLG+ε, where EMLG is the energy spectrum of a monolayer graphene and ε is controlled by the magnitude of electric field and layer number. The associated wave functions are clearly specified. Such a method is applied to analytically study the combined effect of the vertical electric field and magnetic fields on LLs and wave functions in AA-stacking N-layer graphenes. An analytical formula of the gate-tuned LLs clearly exhibits the dependence of LLs on the number of layers, strength of gated field and magnetic fields.

Dynamic effects of electromagnetic wave on a damped two-level atom: Exact solution via path integral

The spin coherent state path integral describing the dynamics of a two-level system interacting with electromagnetic wave of circular polarization is considered. The propagator is first written in the standard form by replacing the spin by a unit vector aligned along the polar and azimuthal directions. Then it is determined exactly using perturbation methods. Thus, the exact energy spectra with corresponding wave functions are deduced.

Classical Energy Spectrum of the Hirota Nonlinear Oscillator

The exact solutions and classical energy spectra for the one-spring and two-spring models of the Hirota anharmonic oscillator have been obtained. The dependences of nonlinear oscillations frequencies on energy have been found. It was shown that the equation of the two-spring model of the Hirota nonlinear oscillator can be formally derived from the equation of Duffing oscillator. The model of the Hirota anharmonic lattice is modified by considering the dissipation and external homogeneous alternating force.

Compensational scintillation detector with a flat energy response for flash X-ray measurements

To measure the intensity of flash X-ray sources directly, a novel scintillation detector with a fast time response and flat energy response is developed by combining film scintillators of doped ZnO crystal and fast organic scintillator together. Through compensation design, the dual-scintillator detector (DSD) achieved a flat energy response to X-rays from tens of keV to several MeV, and sub-nanosecond time response by coupling to ultrafast photo-electronic devices. A prototype detector was fabricated according to the theoretical design; it employed ZnO:In and EJ228 with thicknesses of 0.3 mm and 0.1 mm, respectively. The energy response of this detector was tested on monoenergetic X-ray and γ-ray sources. The detector performs very well with a sensitivity fluctuation below 5% for 8 discrete energy points within the 40-250 keV energy region and for other energies of 662 keV and 1.25 MeV as well, showing good accordance with the theoretical design. Additionally, the detector works properly for the application to the flash X-ray radiation field absolute intensity measurement. This DSD may be very useful for the diagnosis of time-resolved dynamic physical processes of flash X-ray sources without knowing the exact energy spectrum.

A note on the one-dimensional hydrogen atom with minimal length uncertainty

We present the exact energy spectrum and eigenfunctions of the one-dimensional hydrogen atom in the presence of the minimal length uncertainty. By requiring the self-adjointness property of the Hamiltonian, we completely determine the quantization condition. We indicate that the single-valuedness criterion of the eigenfunctions in a non-deformed case is an emergent condition and the semiclassical solutions exactly coincide with the quantum mechanical results. The behavior of the wavefunctions at the origin in coordinate space and in quasiposition space is discussed finally.

Spectral properties of quantum dots influenced by a confining potential model

We obtain the exact energy spectra and corresponding wave functions of the spherical quantum dots for any (n,l) state in the presence of a combination of pseudo-harmonic, Coulomb and linear confining potential terms within the exact analytical iteration method (EAIM). The interaction potential model under consideration is labeled as the Cornell modified-plus-harmonic (CMpH) type which is a correction form to the harmonic, Coulomb and linear confining potential terms.

Two-dimensional Born-Infeld gauge theory: Spectrum, string picture, and large-Nphase transition

We analyze $U(N)$ Born-Infeld gauge theory in two spacetime dimensions. We derive the exact energy spectrum on the circle and show that it reduces to $N$ relativistic fermions on a dual space. This contrasts to the Yang-Mills case that reduces to nonrelativistic fermions. The theory admits a string theory interpretation, analogous to the one for ordinary Yang-Mills, but with higher-order string interactions. We also demonstrate that the partition function on the sphere exhibits a large-$N$ phase transition in the area and calculate the critical area. The limit in which the dimensionless coupling of the theory goes to zero corresponds to massless fermions, admits a perturbatively exact free string interpretation and exhibits no phase transition.

Extension of Bogoliubov theory for a many-body system with a time-scale hierarchy: The quantum mechanics of second Josephson oscillations

Adiabatic approximations are a powerful tool for simplifying nonlinear quantum dynamics, and are applicable whenever a system exhibits a hierarchy of time scales. Current interest in small nonlinear quantum systems, such as few-mode Bose-Hubbard models, warrants further development of adiabatic methods in the particular context of these models. Here we extend our recent work on a simple four-mode Bose-Hubbard model with two distinct dynamical time scales, in which we showed that among the perturbations around excited stationary states of the system is a slow collective excitation that is not present in the Bogoliubov spectrum. We characterized this mode as a resonant energy exchange with its frequency shifted by nonlinear effects, and referred to it as a second Josephson oscillation, in analogy with the second sound mode of liquid helium II. We now generalize our previous theory beyond the mean field regime, and construct a general Bogoliubov free quasiparticle theory that explicitly respects the system's adiabatic invariant as well the exact conservation of particles. We compare this theory to the numerically exact quantum energy spectrum with up to forty particles, and find good agreement over a significant range of parameter space.

Langer Modification in WKB Quantization for Translationally Shape Invariant Potentials

When the Langer modification is applied to Coulomb potential, the standard WKB quantization yields an exact energy spectrum for the potential. This Langer modification has been known to be related to the centrifugal term appearing in Coulomb potential. But we find that a similar modification exists for all translationally shape invariant potentials without referring to the centrifugal term. The characteristic shape of the potentials accounts for the generalized version of Langer modification that makes the WKB quantization valid for all translationally shape invariant potentials.

EXACT SOLVING q DEFORMED BOSON MODEL UNDER OPEN BOUNDARY CONDITION

Utilizing the algebraic Bethe–Ansatz method, the Hamiltonian of q deformed boson model and its eigenvalue equation are calculated under the integrable open boundary condition. Rely on them, we give the exact energy spectrum and discuss two limit cases of the model.

Novel bound states treatment of the two dimensional Schrödinger equation with pseudocentral potentials plus multiparameter noncentral potential

By converting the rectangular basis potential V(x,y) into the form as V(r)+V(r, phi) described by the pseudo central plus noncentral potential, particular solutions of the two dimensional Schrodinger equation in plane-polar coordinates have been carried out through the analytic approaching technique of the Nikiforov and Uvarov (NUT). Both the exact bound state energy spectra and the corresponding bound state wavefunctions of the complete system are determined explicitly and in closed forms. Our presented results are identical to those of the previous works and they may also be useful for investigation and analysis of structural characteristics in a variety of quantum systems

Bound-states of the [formula omitted]-dimensional DKP equation with a pseudoscalar linear plus Coulomb-like potential

We solve the DKP equation with a pseudoscalar linear plus Coulomb-like potential in a two-dimensional space–time. The problem is mapped into an effective quadratic plus inversely quadratic potential in a Sturm–Liouville type equation. The exact energy spectrum and the corresponding normalized eigenstates are obtained. The particular case of the DKP oscillator is also discussed.

Relative velocities of solids in a turbulent protoplanetary disc

We use magnetohydrodynamic simulations to measure relative speeds of solids in a protoplanetary disc with turbulence generated by the magnetorotational instability. Relative velocities are calculated as functions of particle Stokes number St, which measures the aerodynamic coupling to the gas. When relative velocities V rel are calculated between two particles i and j such that St i » St j and St j « 1, the data matches the analytical model of Ormel & Cuzzi. However, if V rel corresponds to two particles with the same St, only the data for the more loosely coupled solids (i.e. those with large St) follow the model. The discrepancy at the low-St end can be attributed to: (i) the numerical disc model's coarse resolution, which is unable to probe smaller turbulent eddies and, therefore, the dominant contribution to the particle relative velocities is given by the interpolation of the gas velocity inside the grid cells; (ii) the sparse particle sampling, which prevents the measurement of relative velocities between two particles in the same place at the same time. The distribution of turbulence-induced relative speeds can have a wide spread of values, which may lead to particle shattering, subject to the turbulent gas velocity. Codes such as the one used in this work, in general, underestimate relative velocities in turbulence for particles with St ≈ 1 because they lack energy on short time-scales (relative to a Kolmogorov spectrum). In making comparisons with theory, it is important to use the exact numerical energy spectrum instead of assuming a Kolmogorov inertial range.

New representation of the multimode phase shifting operator and its application

Based on the rotation transformation in phase space and the technique of integration within an ordered product of operators, the coherent state representation of the multimode phase shifting operator and one of its new applications in quantum mechanics are given. It is proved that the coherent state is a natural language for describing the phase shifting operator or multimode phase shifting operator. The multimode phase shifting operator is also a useful tool to solve the dynamic problems of the multimode coordinate–momentum coupled harmonic oscillators. The exact energy spectra and eigenstates of such multimode coupled harmonic oscillators can be easily obtained by using the multimode phase shifting operator.

Path integral approach to the D-dimensional harmonic oscillator with minimal length

We solve the momentum space path integral for the D-dimensional harmonic oscillator in the context of some deformed commutation relations leading to isotropic nonzero minimal uncertainty in position coordinates. The exact energy spectrum and the corresponding normalized radial momentum space eigenfunctions are obtained from the spectral decomposition of the radial transition amplitude.