Spectrum Of Harmonic Oscillator Research Articles

Spectral Analysis of Proton Eigenfunctions in Crystalline Environments

The Schrödinger equation and Bloch theorem are applied to examine a system of protons confined within a periodic potential, accounting for deviations from ideal harmonic behavior due to real-world conditions like truncated and non-quadratic potentials, in both one-dimensional and three-dimensional scenarios. Numerical computation of the energy spectrum of bound eigenfunctions in both cases reveals intriguing structures, including bound states with degeneracy matching the site number Nw, reminiscent of a finite harmonic oscillator spectrum. In contrast to electronic energy bands, the proton system displays a greater number of possible bound states due to the significant mass of protons. Extending previous research, this study rigorously determines the constraints on the energy gap and oscillation amplitude of the previously identified coherent states. The deviations in energy level spacing identified in the computed spectrum, leading to the minor splitting of electromagnetic modes, are analyzed and found not to hinder the onset of coherence. Finally, a more precise value of the energy gap is determined for the proton coherent states, ensuring their stability against thermal decoherence up to the melting temperature of the hosting metal.

Effect of electric and magnetic field on thermal property of two dimensional harmonic oscillator

The energy spectrum of a 2D quantum harmonic oscillator confined by a hard wall circular cavity of size r0 in the presence of static electric and the magnetic field is computed numerically using the Bessel function basis set and the linear variational process. The effect of the size of the circular cavity, magnetic field, and electric field strength is studied on the energy levels, the radial transition matrix elements, and the thermal properties of the system like specific heat, Helmholtz free energy, and entropy using canonical partition function. Blueshift is observed for an increase in magnetic and electric field strength. The harmonic oscillator problem is critical for understanding any natural system that undergoes deformation and performs simple harmonic motion.

Quantum-Mechanical Explicit Solution for the Confined Harmonic Oscillator Model with the Von Roos Kinetic Energy Operator

Exactly-solvable confined model of the nonrelativistic quantum harmonic oscillator is proposed. Its position-dependent effective mass Hamiltonian is defined via the von Roos kinetic energy operator. The confinement effect to harmonic oscillator potential is included as a result of certain behaviour of the position-dependent effective mass. The corresponding Schrodinger equation in the canonical approach is exactly solved and it is shown that the discrete energy spectrum of the system under consideration depends on the confinement parameter a, von Roos parameters α, β, γ and has a non-equidistant form. Wave functions of the stationary states of the model are expressed through the Gegenbauer polynomials. The limit a → ∞ recovers both equidistant energy spectrum and wave functions of the stationary nonrelativistic harmonic oscillator expressed by Hermite polynomials.

Effect of noncommutativity on the spectrum of free particle and harmonic oscillator in rotationally invariant noncommutative phase space

We consider rotationally invariant noncommutative algebra with tensors of noncommutativity constructed with the help of additional coordinates and momenta. The algebra is equivalent to the well-known noncommutative algebra of canonical type. In the noncommutative phase space, rotational symmetry influence of noncommutativity on the spectrum of free particle and the spectrum of harmonic oscillator is studied up to the second-order in the parameters of noncommutativity. We find that because of momentum noncommutativity, the spectrum of free particle is discrete and corresponds to the spectrum of harmonic oscillator in the ordinary space (space with commutative coordinates and commutative momenta). We obtain the spectrum of the harmonic oscillator in the rotationally invariant noncommutative phase space and conclude that noncommutativity of coordinates affects its mass. The frequency of the oscillator is affected by the coordinate noncommutativity and the momentum noncommutativity. On the basis of the results, the eigenvalues of squared length operator are found and restrictions on the value of length in noncommutative phase space with rotational symmetry are analyzed.

Importance of Vibronic Effects in the UV-Vis Spectrum of the 7,7,8,8-Tetracyanoquinodimethane Anion.

We present a computational method for simulating vibronic absorption spectra in the ultraviolet-visible (UV-vis) range and apply it to the 7,7,8,8-tetracyanoquinodimethane anion (TCNQ-), which has been used as a ligand in black absorbers. Gaussian broadening of vertical electronic excitation energies of TCNQ- from linear-response time-dependent density functional theory produces only one band, which is qualitatively incorrect. Thus, the harmonic vibrational modes of the two lowest doublet states were computed, and the vibronic UV-vis spectrum was simulated using the displaced harmonic oscillator approximation, the frequency-shifted harmonic oscillator approximation, and the full Duschinsky formalism. An efficient real-time generating function method was implemented to avoid the exponential complexity of conventional Franck-Condon approaches to vibronic spectra. The obtained UV-vis spectra for TCNQ- agree well with experiment; the Duschinsky rotation is found to have only a minor effect on the spectrum. Born-Oppenheimer molecular dynamics simulations combined with calculations of the electronic excitation energies for a large number of molecular structures were also used for simulating the UV-vis spectrum. The Born-Oppenheimer molecular dynamics simulations yield a broadening of the energetically lowest peak in the absorption spectrum, but additional vibrational bands present in the experimental and simulated quantum harmonic oscillator spectra are not observed in the molecular dynamics simulations. Our results underline the importance of vibronic effects for the UV-vis spectrum of TCNQ-, and they establish an efficient method for obtaining vibronic spectra using a combination of linear-response time-dependent density functional theory and a real-time generating function approach.

Energy spectrum inverse problem of q-deformed harmonic oscillator and WBK approximation

Using the connection between q-deformed harmonic oscillator and Morse-like anharmonic potential we investigate the energy spectrum inverse problem. Consider some energy levels of energy spectrum of q-deformed harmonic oscillator are known, we construct the corresponding Morse-like potential then find out the deform parameter q. The application possibility of using the WKB approximation in the energy spectrum inverse problem was discussed for the cases of parabolic potential (harmonic oscillator), Morse-like potential (q-deformed harmonic oscillator). so we consider our deformed-three-levels simple model, where the set-parameters of Morse potential and the corresponding set-parameters of level deformations are easily and explicitly defined. For practical problems, we propose the deformed- three-levels simple model, where the set-parameters of Morse potential and the corresponding set-parameters of level deformations are easily and explicitly defined.

Landau levels and Shubnikov–de Haas oscillations in monolayer transition metal dichalcogenide semiconductors

We study the Landau level (LL) spectrum using a multi-band theory in monolayer transition metal dichalcogenide semiconductors. We find that in a wide magnetic field range the LL can be characterized by a harmonic oscillator spectrum and a linear-in-magnetic field term which describes the valley degeneracy breaking. The effect of the non-parabolicity of the band-dispersion on the LL spectrum is also discussed. Motivated by recent magnetotransport experiments, we use the self-consistent Born approximation and the Kubo formalism to calculate the Shubnikov–de Haas oscillations of the longitudinal conductivity. We investigate how the doping level, the spin-splitting of the bands and the broken valley degeneracy of the LLs affect the magnetoconductance oscillations. We consider monolayer MoS2 and WSe2 as concrete examples and compare the results of numerical calculations and an analytical formula which is valid in the semiclassical regime. Finally, we briefly analyze the recent experimental results (Cui et al 2015 Nat. Nanotechnol. 10 534) using the theoretical approach we have developed.

An xp model on AdS2 spacetime

In this paper we formulate the xp model on the AdS2 spacetime. We find that the spectrum of the Hamiltonian has positive and negative eigenvalues, whose absolute values are given by a harmonic oscillator spectrum, which in turn coincides with that of a massive Dirac fermion in AdS2. We extend this result to generic xp models which are shown to be equivalent to a massive Dirac fermion on spacetimes whose metric depend of the xp Hamiltonian. Finally, we construct the generators of the isometry group SO(2,1) of the AdS2 spacetime, and discuss the relation with conformal quantum mechanics.

Quantizations from reproducing kernel spaces

The purpose of this work is to explore the existence and properties of reproducing kernel Hilbert subspaces of L2(C,d2z/π) based on subsets of complex Hermite polynomials. The resulting coherent states (CS) form a family depending on a nonnegative parameter s. We examine some interesting issues, mainly related to CS quantization, like the existence of the usual harmonic oscillator spectrum despite the absence of canonical commutation rules. The question of mathematical and physical equivalences between the s-dependent quantizations is also considered.

A Classical Explanation of Quantization

In the context of our recently developed emergent quantum mechanics, and, in particular, based on an assumed sub-quantum thermodynamics, the necessity of energy quantization as originally postulated by Max Planck is explained by means of purely classical physics. Moreover, under the same premises, also the energy spectrum of the quantum mechanical harmonic oscillator is derived. Essentially, Planck’s constant h is shown to be indicative of a particle’s “zitterbewegung” and thus of a fundamental angular momentum. The latter is identified with quantum mechanical spin, a residue of which is thus present even in the non-relativistic Schrödinger theory.

ENTANGLED STATE REPRESENTATIONS IN NONCOMMUTATIVE PHASE SPACE

The entangled state representation has been constructed on noncommutative phase space. Using this appropriate representation, the energy spectrum of general two-dimensional harmonic oscillator has been obtained exactly.

Asymptotics of spectrum of a harmonic oscillator perturbed by a nonsmooth potential

1. In this paper we study the operator H = H0 + V in L2(R), where H0 = −d2/dx2 + x2, V is the operator of multiplication by a real, measurable, decreasing for x → ∞ function. It is well-known (see, e.g., [1], p. 326) that the spectrum of the operator H0 consists of the numbers 2n+ 1, and the corresponding normalized eigenfunctions are φn(x) = Hn(x)e−x 2/2/ √ 2nn! √ π, n = 0, 1, 2, . . . , where Hn(x) are the Chebyshev–Hermite polynomials. The asymptotics of eigenvalues of the perturbed operator H = H0 + V for smooth decreasing at infinity functions was first studied in detail in [2], where an etalon solution was obtained with the help of the Airy functions ([3], p. 377). Since V (x) is not necessarily smooth, one cannot immediately apply the method of etalon solutions to the function q(x) = x2 + V (x). In this paper we use the apparatus of perturbation theory based on the study of the asymptotic representation of the kernel of the resolvent of the unperturbed operator. Let λn stand for eigenvalues of the operator H0, let Pn denote the corresponding projectors onto eigensubspaces, and let R0(λ) be the resolvent of the operator H0, R0(λ) = (H0 − λ)−1. According to [4], if V satisfies the condition

Parametric Resonance of Optically Trapped Aerosols

The Brownian dynamics of an optically trapped water droplet are investigated across the transition from over- to underdamped oscillations. The spectrum of position fluctuations evolves from a Lorentzian shape typical of overdamped systems (beads in liquid solvents) to a damped harmonic oscillator spectrum showing a resonance peak. In this later underdamped regime, we excite parametric resonance by periodically modulating the trapping power at twice the resonant frequency. The power spectra of position fluctuations are in excellent agreement with the obtained analytical solutions of a parametrically modulated Langevin equation.

Harmonic oscillator spectrum for glueball states in 2 + 1 dimensions in the large N limit

We compute the energies of the glueball states in SU(N) field theory in 2 + 1 dimensions using Hamiltonian techniques. Using a new result which gives the integrals required analytically we are able to calculate for N ≤ 25, and find that the spectrum of the excited states, extrapolated to the N → ∞, is of the harmonic oscillator type. © 2004 Published by Elsevier B.V.

SU(N) for large N in the Hamiltonian formalism in 2+1 dimensions

We describe how to obtain improved Hamiltonians for pure glue SU(N) gauge theory, and how to analytically estimate the excited state energies in 2+1 dimensions for any N . We find that the N → ∞ excited states appear to follow a 2 dimensional harmonic oscillator spectrum.

Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework

In the context of a two-parameter (α, β) deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using an extension of the techniques of conventional supersymmetric quantum mechanics (SUSYQM) combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters α, β vanishes, in which case shape invariance under parameter translation occurs. In the special case where α = β ≠ 0, the oscillator Hamiltonian is shown to coincide with that of the q-deformed oscillator with q > 1 and its eigenvectors are therefore n-q-boson states. In the general case where 0 ≠ α ≠ β ≠ 0, the eigenvectors are constructed as linear combinations of n-q-boson states by resorting to a Bargmann representation of the latter and to q-differential calculus. They are finally expressed in terms of a q-exponential and little q-Jacobi polynomials.

QUANTUM FIELD THEORY AND COMPUTATIONAL PARADIGMS

We introduce the basic theory of quantization of radiation field in quantum physics and explain how it relates to the theory of recursive functions in computer science. We outline the basic differences between quantum mechanics (QM) and quantum field theory (QFT) and explain why QFT is better suited for a computational paradigm — based on algorithmic requirement, countably infinite degrees of freedom and the creation of macroscopic output objects. The quanta of the radiation field correspond to the non-negative integers and the harmonic oscillator spectra correspond to the recursive computation — with the creation and annihilation operators, respectively, playing the same role as the successor and predecessor in computability theory. Accordingly, this approach relates the classical computational model and the quantum physical model more directly than the Turing machine approach used earlier. Also, the application of Lambda calculus formalism and the associated denotational semantics (that is widely used in the classical computational paradigm involving recursive functions) for applications to computational paradigm based on quantum field theory is described. Finally, we explain where QFT and conventional paradigm depart from each other, and examine the concept of fixed points, phase transitions, programmability, emergent computation and related open problems.

Dissipation and quantization

We show that the dissipation term in the Hamiltonian for a couple of classical damped–amplified oscillators manifests itself as a geometric phase and is actually responsible for the appearance of the zero point energy in the quantum spectrum of the 1D linear harmonic oscillator. We also discuss the thermodynamical features of the system. Our work has been inspired by 't Hooft proposal according to which information loss in certain classical systems may lead to “an apparent quantization of the orbits which resembles the quantum structure seen in the real world”.

Non-local potentials and rotational bands of resonances in ion collisions

Sequences of rotational resonances (rotational bands) and corresponding antiresonances are observed in ion collisions. In this paper we propose a description which combines collective and single-particle features of cluster collisions. It is shown how rotational bands emerge in many-body dynamics, when the degeneracies proper of the harmonic oscillator spectra are removed by adding interactions depending on the angular momentum. These interactions can be properly introduced in connection with the exchange forces and the antisymmetrization, and give rise to a class of non-local potentials whose spectral properties are analyzed in detail. In particular, we give a classification of the singularities of the resolvent, which are associated with bound states and resonances. The latter are then studied using an appropriate type of collective coordinates, and a hydrodynamical model of the trapping, responsible for the resonances, is then proposed. Accordingly, we derive, from the uncertainty principle, a spin-width of the unstable states which can be related to their angular lifetime.

On aq-analogue of the spin-orbit coupling

Based on the tensor method, a q-analogue of the spin-orbit coupling is introduced in a q-deformed Schrodinger equation, previously derived for a central potential. Analytic expressions for the matrix elements of the representations j = l±½ are derived. The spectra of the harmonic oscillator and the Coulomb potential are calculated numerically as a function of the deformation parameter, without and with the spin-orbit coupling. The harmonic oscillator spectrum presents strong analogies with the bound spectrum of a Woods-Saxon potential customarily used in nuclear physics. The Coulomb spectrum simulates relativistic effects. The addition of the spin-orbit coupling reinforces this picture.