On the Relativistic Harmonic Oscillator

The relativistic harmonic oscillator represents a unique energy-conserving oscillatory system. The detailed characteristics of the solution of this oscillator are displayed in both weak- and extreme-relativistic limits using different expansion procedures, for each limit. In the weak-relativistic limit, a Normal Form expansion is developed, which yields an approximation to the solution that is significantly better than in traditional asymptotic expansion procedures. In the extreme-relativistic limit, an expansion of the solution in terms of a small parameter that measures the proximity to the limit (v/c) →1 yields an excellent approximation for the solution throughout the whole period of oscillations. The variation of the coefficients of the Fourier expansion of the solution from the weak- to the extreme-relativistic limits is displayed.

Loop Quantum Gravity Harmonic Oscillator

Attempts to unify Gravity Theory and Quantum Field Theory (QFT) under Loop Quantum Gravity Theory (LQG), are diverse;a dividing line between classical and quantum is sought with Schrödinger cat-state experiments. A Primordial Field Theory-based alternative is presented, and a gravity-based harmonic oscillator developed. With quantum theory applied at micro-scales and gravity theory at meso- and macro-scales, this scale-gap contributes to the conceptual problems associated with Loop Quantum Gravity. Primordial field theory, spanning all scales, is used to conceptually stretch key ideas across this gap. An LQG interpretation of the wave function associated with the oscillator is explained.

A Model Effective Mass Quantum Anharmonic Oscillator and Its Thermodynamic Characterization

Using a model anharmonic oscillator with asymptotically decreasing effective mass to study the effect of compositional grading on the quantum mechanical properties of a semiconductor heterostructure, we determine the exact bound states and spectral values of the system. Furthermore, we show that ordering ambiguity only brings about a spectral shift on the quantum anharmonic oscillator with spatially varying effective mass. A study of thermodynamic properties of the system reveals a resonance condition dependent on the magnitude of the anharmonicity parameter. This resonance condition is seen to set a critical value on the said parameter beyond which a complex valued entropy which is discussed, emerges.

Sampled-data synchronization of coupled harmonic oscillators with controller failure and communication delays

In this letter, a distributed protocol for sampled-data synchronization of coupled harmonic oscillators with controller failure and communication delays is proposed, and a brief procedure of convergence analysis for such algorithm over undirected connected graphs is provided. Furthermore, a simple yet generic criterion is also presented to guarantee synchronized oscillatory motions in coupled harmonic oscillators. Subsequently, the simulation results are worked out to demonstrate the efficiency and feasibility of the theoretical results.

Noncovariant Lagrangians Are Presented Which Yield Two-Component Equations of Motion for a Class of Relativistic Mechanical Systems in 1 + 1 Dimensions Including the Harmonic Oscillator

First, a Lagrangian is presented and authenticated for a Relativistic Harmonic Oscillator in 1 + 1 dimensions. It yields a two-component set of equations of motion. The time-component is the missing piece in all previous discussions of this system! The second result is that this Oscillator Langrangian generalizes to Langrangians for a class of particles in 1 + 1 dimensions subject to an arbitrary potential V which is space dependent only.

Stochastic Resonance in a Harmonic Oscillator Fluctuating Intrinsic Frequency by Asymmetric Dichotomous Noise

The stochastic resonance phenomenon in a harmonic oscillator with fluctuating intrinsic frequency by asymmetric dichotomous noise is investigated in this paper. By using the random average method and Shapiro- Loginov formula, the exact solution of the average output amplitude gain （OAG） is obtained. Numerical results show that OAG depends non-monotonically on the noise characteristics： intensity, correlation time and asymmetry. The maximum OAG can be achieved by tuning the noise asymmetry and or the noise correlation time.

Collective stochastic resonance behaviors of two coupled harmonic oscillators driven by dichotomous fluctuating frequency

The collective behaviors of two coupled harmonic oscillators with dichotomous fluctuating frequency are investigated,including stability, synchronization, and stochastic resonance(SR). First, the synchronization condition of the system is obtained. When this condition is satisfied, the mean-field behavior is consistent with any single particle behavior in the system. On this basis, the stability condition and the exact steady-state solution of the system are derived. Comparative analysis shows that, the stability condition is stronger than the synchronization condition, that is to say, when the stability condition is satisfied, the system is both synchronous and stable. Simulation analysis indicates that increasing the coupling strength will reduce the synchronization time. In weak coupling region, there is an optimal coupling strength that maximizes the output amplitude gain(OAG), thus the coupling-induced SR behavior occurs. In strong coupling region, the two particles are bounded as a whole, so that the coupling effect gradually disappears.

Connes distance of 2D harmonic oscillators in quantum phase space

We study the Connes distance of quantum states of two-dimensional(2D)harmonic oscillators in phase space.Using the Hilbert–Schmidt operatorial formulation,we construct a boson Fock space and a quantum Hilbert space,and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional(4D)quantum phase space.Based on the ball condition,we obtain some constraint relations about the optimal elements.We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators.We prove that these two-dimensional distances satisfy the Pythagoras theorem.These results are significant for the study of geometric structures of noncommutative spaces,and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.

The relativistic bound states for a new ring-shaped harmonic oscillator

In this paper a new ring-shaped harmonic oscillator for spin 1/2 particles is studied, and the corresponding eigenfunctions and eigenenergies are obtained by solving the Dirac equation with equal mixture of vector and scalar potentials. Several particular cases such as the ring-shaped non-spherical harmonic oscillator, the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.

Quantization of the 1-D Forced Harmonic Oscillator in the Space (<i>x</i>, <i>v</i>)

The quantization of the forced harmonic oscillator is studied with the quantum variable (x, v∧), with the commutation relation , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion K (x, v, t) of the classical system, The comparison with the quantization in the space (x, p) is done with the usual Schrödinger’s equation for the Hamiltonian H(x, p, t), and with the commutation relation . It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (x, v∧) quantization has fewer oscillations than the (x, p∧) quantization, the average energy of the system is higher in (x, p∧) quantization than on the (x, v∧) quantization, and the Boltzmann-Shannon entropy on the (x, p∧) quantization is higher than on the (x, v∧) quantization.

Coherent States and Uncertainty Product of the Harmonic Oscillator with Position-Dependent Mass

In this work, we applied the invariant method to calculate the coherent state of the harmonic oscillator with position-dependent mass, which in modern physics has great application. We also obtain the calculation of Heisenberg’s uncertainty principle, and we will show that it is verified.

Quantum Mechanical Path Integral in Phase Space and Class of Harmonic Oscillators with Varied Frequencies

We present the problem of the time-dependent Harmonic oscillator with time-dependent mass and frequency in phase space and by using a canonical transformation and delta functional integration we could find the propagator related to the system. New examples of time-dependent frequencies are presented.

New Possibilities of Harmonic Oscillator Basis Application for Quantum System Description.Two Particles with Coulomb Interaction

This paper is addressed to the problem of Galilei invariant basis construction for identical fermions systems. The recently introduced method for spurious state elimination from expansions in harmonic oscillator basis [1] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310034003400340037003600360038000000 is adopted and applied to bound states of two particles system with Coulomb potential description. Traditional expansions in this case demonstrate the extremely well-known slow convergence, and hence this is the best problem with known exact solutions for the test of the method. Obtained results demonstrate the significant simplification of the problem and fast convergence of expansions. We show that the application of this general method is very efficient in a test case of the energy spectrum calculation problem of two particles with different masses interacting with Coulomb potential.

Response Solutions for Degenerate Reversible Harmonic Oscillators with Zero-average Perturbation

In this paper,we consider a class of normally degenerate quasi-periodically forced reversible systems,obtained as perturbations of a set of harmonic oscillators,{x˙=y+∈f1(ωt,x,y),y˙=λx^(l)+∈f2(ωt,x,y),where 0≠λ∈R,l>1 is an integer and the corresponding involution G is(−θ,x,−y)→(θ,x,y).The existence of response solutions of the above reversible systems has already been proved in[22]if[f2(ωt,0,0)]satisfies some non-zero average conditions(See the condition(H)in[22]),here[·]denotes the average of a continuous function on T^(d).However,discussing the existence of response solutions for the above systems encounters difficulties when[f_(2)(ωt,0,0)]=0,due to a degenerate implicit function must be solved.This article will be doing work in this direction.The purpose of this paper is to consider the case where[f2(ωt,0,0)]=0.More precisely,with 2p<l,if f_(2)satisfies[f_(2)(ωt,0,0)]=[∂f_(2)(ωt,0,0)/∂x]=[∂^(2)f_(2)(ωt,0,0)/∂x2]=···=[∂p−1f2(ωt,0,0)∂xp−1]=0,eitherλ−1[∂pf2(ωt,0,0)∂xp]<0 as l−p is even orλ−1[∂pf2(ωt,0,0)∂xp]=0 as l−p is odd,we obtain the following results:(1)Forλ>˜0(seeλ˜in(2.2))and sufficiently small,response solutions exist for eachωsatisfying a weak non-resonant condition;(2)Forλ<˜0 and∗sufficiently small,there exists a Cantor set E∈(0,∗)with almost full Lebesgue measure such that response solutions exist for each∈E ifωsatisfies a Diophantine condition.In the remaining case whereλ−1[∂pf2(ωt,0,0)∂xp]>0 and l−p is even,we prove the system admits no response solutions in most regions.

Demonstration and operation of quantum harmonic oscillators in an AlGaAs–GaAs heterostructure

The quantum harmonic oscillator(QHO),one of the most important and ubiquitous model systems in quantum mechanics,features equally spaced energy levels or eigenstates.Here we present a new class of nearly ideal QHOs formed by hydrogenic substitutional dopants in an AlGaAs/GaAs heterostructure.On the basis of model calculations,we demonstrate that,when aδ-doping Si donor substitutes the Ga/Al lattice site close to AlGaAs/GaAs heterointerface,a hydrogenic Si QHO,characterized by a restoring Coulomb force producing square law harmonic potential,is formed.This gives rise to QHO states with energy spacing of~8–9 meV.We experimentally confirm this proposal by utilizing gate tuning and measuring QHO states using an aluminum single-electron transistor(SET).A sharp and fast oscillation with period of~7–8 mV appears in addition to the regular Coulomb blockade(CB)oscillation with much larger period,for positive gate biases above 0.5 V.The observation of fast oscillation and its behavior is quantitatively consistent with our theoretical result,manifesting the harmonic motion of electrons from the QHO.Our results might establish a general principle to design,construct and manipulate QHOs in semiconductor heterostructures,opening future possibilities for their quantum applications.

Exact solution to two-dimensional isotropic charged harmonic oscillator in uniform magnetic field in non-commutative phase space

In this paper, the isotropic charged harmonic oscillator in uniform magnetic field is researched in the non-commutative phase space; the corresponding exact energy is obtained, and the analytic eigenfunction is presented in terms of the confluent hypergeometric function. It is shown that in the non-commutative space, the isotropic charged harmonic oscillator in uniform magnetic field has the similar behaviors to the Landau problem.

Generalized path-integral solution and coherent states of the harmonic oscillator in D-dimensions

We construct a general form of propagator in arbitrary dimensions and give an exact wavefunction of a time- dependent forced harmonic oscillator in D（D ≥ 1） dimensions. The coherent states, defined as the eigenstates of annihilation operator, of the D-dimensional harmonic oscillator are derived. These coherent states correspond to the minimum uncertainty states and the relation between them is investigated.

The fractional features of a harmonic oscillator with position-dependent mass

In this study,a harmonic oscillator with position-dependent mass is investigated.Firstly,as an introduction,we give a full description of the system by constructing its classical Lagrangian;thereupon,we derive the related classical equations of motion such as the classical Euler–Lagrange equations.Secondly,we fractionalize the classical Lagrangian of the system,and then we obtain the corresponding fractional Euler–Lagrange equations(FELEs).As a final step,we give the numerical simulations corresponding to the FELEs within different fractional operators.Numerical results based on the Caputo and the Atangana-Baleanu-Caputo(ABC)fractional derivatives are given to verify the theoretical analysis.

Multiquark cluster form factors in the relativistic harmonic oscillator model

A QCD multiquark cluster system is studied in the relativistic harmonic oscillator potential model （RHOPM）, and the electromagnetic form factors of the pion, proton and deuteron in the RHOPM are predicted. The calculated theoretical results are then compared with existing experimental data, finding very good agreement between the theoretical predictions and experimental data for these three target particles. We claim that this model can be applied to study QCD hadronic properties, particularly neutron properties, and to find six-quark cluster and/or nine-quark cluster probabilities in light nuclei such as helium 3He and tritium 3H. This is a problem of particular importance and interest in quark nuclear physics.

Observation about the Classical Electromagnetic Gauge Transformation and Its Quantum Correspondence

Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A, we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B=∇×A if and only if B≠0.