Energy Eigenvalues Research Articles

Energy spectrum of selected diatomic molecules (H2, CO, I2, NO) by the resolution of Schrodinger equation for combined potentials via NUFA method.

To determine the properties of diatomic molecules, studying their chemical bond energy spectrum is essential since it enables the assessment of their characteristics. This research presents diatomic molecules spectroscopic characteristics and rovibrational energy (H2, CO, I2, NO). The Schrodinger equation is solved to determine these energies, considering the presence of a combination of two distinct potentials. the inverse quadratic Yukawa potential in combination with the screened modified Kratzer. This work used the Greene-Aldrich assumption and the Nikiforov-Uvarov functional analysis approach as analytical tools to solve the Schrodinger equation and determine the energy spectrum of diatomic molecules (H2, CO, I2, NO). The use of Mathematica software allows for the calculation of the eigenvalues of energy of the previously mentioned diatomic molecules (H2, CO, I2, NO) based on their rovibrational energies in the final equation. By comparing the eigenvalue findings with previous research, it was seen that the technique yielded the expected and desirable outcomes.

Combined effect of rashba spin-orbit interaction, hydrostatic pressure and temperature on energy dispersion based ballistic conductance of InAs tunnel-coupled (double) quantum wire under exterior magnetic and electric field

In this work the change in energy dispersion curve and ballistic conductance for InAs tunnel-coupled double quantum wire (DQWR) under the influence of the external magnetic field, electric field, hydrostatic pressure, temperature, and Rashba spin-orbit interaction (SOI) are studied theoretically in detail. A constant magnetic field transverse to the fundamental plane is applied to the DQWR. The Landuer-Büttiker formalism is used to determine the transport characteristics, and the energy eigenvectors and eigenvalues are derived within the parameters of the diagonalization approach. The numerical results indicate that the energy spectra of the system are influenced by external factors such as magnetic field, electric field, temperature, hydrostatic pressure, and Rashba SOI, resulting in variations in lateral/vertical and downward/upward shifts. These factors also cause oscillatory patterns in ballistic conductance due to the oddity of energy subbands. These results have significant implications for the advancement of double-quantum wire-based devices.

On a family of finitely many point interaction Hamiltonians free of ultraviolet pathologies

Abstract We re-investigate the entire family of many center point interaction Hamiltonians. Under the assumption of exchange symmetry with respect to the point positions, we show that a large sub-family of point interaction Hamiltonian operators does not become either singular or trivial when the positions of two or more scattering centers tend to coincide. In this sense, they appear to be renormalized by default as opposed to the point interaction Hamiltonians usually considered in the literature. Functions in their domains satisfy regularized boundary conditions which turn out to be very similar to the ones proposed recently in many-body quantum mechanics to define three-particle system Hamiltonians with contact interactions bounded from below. In the two-center case, we study the behavior of the negative eigenvalues as a function of the center distance. The result is used to analyze a formal Born–Oppenheimer approximation of a three-particle system with two heavy bosons and one light particle. We demonstrate that this simplified model describes a stable system (no ‘fall to the center’ problem is present). Furthermore, in the unitary limit, the energy spectrum is characterized by an infinite sequence of negative energy eigenvalues accumulating at zero according to the geometrical Efimov law.

Degeneracy and Photon Trapping in a Dissipationless Two-Mode Optomechanical Model.

In this work, we theoretically study a finite and undamped two-mode optomechanical model consisting of a high quality optical cavity containing a thin, elastic, and dielectric membrane. The main objective is to investigate the precursors of quantum phase transition in such a model by studying the behavior of some observables in the ground state. By controlling the coupling between membrane and modes, we find that the two lowest energy eigenstates become degenerate, as is indicated by the behavior of the mean value of some operators and by other quantifiers as a function of the coupling. Such degenerate states are characterized by a coherent superposition of eigenstates describing one of the two modes preferentially populated and the membrane dislocated from its equilibrium position due the radiation pressure (Schrödinger's cat states). The delocalization of the compound system photons+membrane results in an increase in fluctuations as measured by Robertson-Schrödinger uncertainty relations.

Energy spectrum and applications of Eckart plus Hellmann potential in hyperspherical coordinates

We investigate the approximate bound-state solutions of the N-dimensional Schrödinger equation with a combination of Eckart and Hellmann potentials via the asymptotic iteration approach. By the use of the Greene-Aldrich approximation, which is valid for small values of the screening parameter, we establish the N-dimensional energy spectrum and the N-dimensional radial wave function in approximate analytic form. In hyperspherical coordinates, the normalized radial wave function is expressed in terms of hypergeometric and Jacobi polynomials. To double-check the energy spectrum in hyperspherical coordinates, we utilize polynomial solutions in view of the asymptotic iteration approach. The expressions for the expectation values of inverse position, square of inverse position, kinetic energy, and square of momentum are derived in hyperspherical coordinates by using the Hellmann-Feynmann theorem. We also present the analytically calculated energy eigenvalues for Eckart plus Hellmann potential. We deduce special forms of the relevant potential, such as Eckart, Hellmann, and Hulthén potentials.

Investigation of linear and nonlinear optical properties in quantum dots: Influence of magnetic and AB flux fields

In the present work, analytical expressions for the energy eigenvalues and eigen functions are obtained by solving the radial Schrödinger equation for inverse quadratic Hellmann plus Yukawa (IQHY) potential using Nikiforov-Uvarov functional analysis method. As an application part of this work, expressions for the linear and third-order nonlinear absorption coefficients and refractive index changes are computed to study the optical properties of the spherical GaAs quantum dots. We extend the investigation of the IQHY potential by incorporating the effects of magnetic and Aharonov–Bohm (AB) flux fields. Our results reveal that the optical properties of the GaAs quantum dots are strongly influenced by variations in the dot radius, potential height and intensity of incident photons. Specifically, we observe that the introduction of magnetic and AB flux fields significantly alters the energy eigenvalues and eigenfunctions, leading to distinct optical characteristics. The results of present study are found in consistent with other theoretical studies available in the literature.

Effects of anisotropy on the electronic structure and nonlinear optical rectification in a quantum dot with hydrogen molecular ion

In this study, the effects of anisotropy and quantum dot size on energy levels and nonlinear optical rectification of a D_{2}^{ + } molecular complex confined to a two-dimensional quantum dot are investigated. The energy eigenvalues and corresponding eigenfunctions of the resulting eigenvalue equation are obtained using the adiabatic approach and the two-dimensional diagonalization method. We conclude that the evolution of the energy spectrum depends predominantly on the anisotropy-induced change of geometric confinement size. In addition, it has been determined that the peak position of the nonlinear optical rectification shifts to lower or higher energies as the asymmetry of the system increases, depending on whether the QD is in the oblate or prolate case. This feature can be considered an advantage in tuning the nonlinear optical properties of systems based on two-dimensional semiconductor heterostructures.

Impact of topological defects and Yukawa potential combined with inverse square on eigenvalue spectra of diatomic molecules O 2, NO, LiH, HCl

In this paper, we investigate a quantum system composed of non-relativistic particles interacting with an external potential while in the presence of a topological defect produced by a point-like global monopole. We derive the radial equation of the Schrödinger equation for this system, considering a Yukawa potential combined with inverse square potential within the backdrop of a point-like global monopole. To determine the eigenvalue solutions for this quantum system, we employ a suitable approximation scheme known as the Pekeris approximation. This approximation is applied to the centrifugal term, and we solve the resulting radial equation using the parametric Nikiforov-Uvarov (NU) method. Subsequently, we examine the quantum system when only Yukawa potential is present, and we obtain the eigenvalue solutions using the same procedure. In both cases, we observe that the presence of the global monopole topological defect has a significant impact on the energy spectrum of quantum particles interacting with an external potential. To illustrate this effect, we consider a few diatomic molecules, such as O 2, NO, LiH, and HCl. We present the energy spectrum and compare our results with previously published findings in the literature. Furthermore, we generate several plots to visually depict the influence of the global monopole on the energy eigenvalues for these different molecules.

Twirling Operations to Produce Energy Eigenstates of a Hamiltonian by Classically Emulated Quantum Simulation

We propose a simple procedure to produce energy eigenstates of a Hamiltonian with discrete eigenvalues. We use ancilla qubits and quantum entanglement to separate an energy eigenstate from the other energy eigenstates. We exhibit a few examples derived from the (1 + 1)‐dimensional massless Schwinger model. Our procedure in principle will be applicable for a Hamiltonian with a finite‐dimensional Hilbert space. Choosing an initial state properly, we can in principle produce any energy eigenstate of the Hamiltonian.

Properties of ground state and quench dynamics of one-dimensional contact repulsive single-spin flipped Fermi gases

Based on the exact solution method, the ground state and quench dynamics properties of one-dimensional single-spin flipped Fermi gas with repulsion interaction are studied. With the Bethe wave function, the single-body correlation function and two-body correlation function of the ground state and those between different eigen-states can be reduced into a summation of simple functions, thereby greatly reducing the computational difficulty. For the system in the ground state, the single-body correlation functions and two-body correlation functions as well as momentum distributions for spin-up particles are investigated in real space with different interaction strengths. As the interaction strength increases, the number of nodes in the single-body correlation function remains unchanged, while the amplitude of oscillation decreases. Meanwhile, the number of peaks in the two-body correlation function increases by one due to interaction, indicating that the spin-down particle behaves as a spin-up particle. The momentum distribution becomes more smooth around Fermi surface with the interaction strength increasing. The interaction quench dynamics is investigated. The system is prepared in the ground state of ideal Fermi gas, and then the interaction strength is quenched to a finite positive value. The system evolves under time-dependent Schrödinger equation. The overlap between the initial state and eigen-state of post-quench interaction strength is expressed in the form of continued multiplication. The square of the modulus of this overlap, which represents the occupation probability, is calculated. We find that the occupation probabilities of the ground state and doubly degenerated excited state always have the first and the second largest value for an arbitrary interaction strength, respectively, which means that the difference in eigenenergy between these two states gives the primary period of oscillation. For relatively large particle number (<inline-formula><tex-math id="M2">\begin{document}$ N\geqslant10$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M2.png"/></alternatives></inline-formula>), the primary period always does not change under different interaction strengths.It is found that in the case of interaction quenching, the momentum distribution and the correlation function show periodic oscillations. When the interaction strength is adjusted to a relatively small value, the oscillation periodicity is well-defined and the oscillation amplitude is small. The system can be approximated by a two-level model. When the interaction strength increases to a very large value, the oscillation periodicity worsens and the amplitude increases, but a primary period remains unchanged. Although the overall deviation is far from the initial state, it is very close to the initial state at time <inline-formula><tex-math id="M3">\begin{document}$ t=mL^2/(2\pi\hbar)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M3.png"/></alternatives></inline-formula>. This is because the difference between most energy eigenvalues is almost an integral multiple of energy unit <inline-formula><tex-math id="M4">\begin{document}$ 2\times\left(2\pi/L\right)^{2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M4.png"/></alternatives></inline-formula>.

Scattering State Solutions of Vector Bosons Interacting with Sun Potential

For vector bosons with spin-1, scattering state solutions have been attained by considering the Duffin-Kemmer-Petiau equation with the Sun interaction field. Based on the obtained solution, relations for phase shift and scattering amplitude have been derived. Furthermore, the bound state energy eigenvalue relation has been derived by taking the scattering amplitude to infinity. The results obtained through the Mathematica software program are presented graphically and numerically. In addition, the effects of the variables in the interaction function on the obtained results are discussed.

A Study on The Optical Properties of Long-Infrared Intraband Transitions of Quadruple Gaas/Alxga1-Xas Quantum Well Under Applied Electric Field

Semiconductor-emitting/absorbing infrared devices are in the common interest of the scientific and industrial community due to their broad application in these fields. GaAs/AlGaAs based devices are one of the most studied semiconductor heterostructures. In this study, I have aimed to design GaAs/AlGaAs quantum well (QW) semiconductor heterostructures to emit/absorb in the long infrared region and studied the optical properties. To do that, I have designed a quadruple QW, which is composed of GaAs/Al0.44Ga0.56As QW and quantum barriers (QB). I have solved the time-independent Schrödinger equation using the finite element method-based matlab code under effective mass approximation. The wave functions and corresponding energy eigenvalues are obtained for varied electric field (EF) intensities. I have shown that our design can operate up to 80 kV/cm, which is the limit for first bounded energy eigenstates. It is observed that E_32 transition provides long-infrared emission/absorption corresponding to the 0.12-0.14 eV transition energy and it is constant with increased EF intensity. In addition, it is seen that the overlap of the wave functions is increasing with EF intensity which enhances radiative transition in the structure. I have calculated the linear absorption coefficient and refractive index change. I have observed that the absorption coefficient of E_32 transition is increasing with EF intensity while E_31 is decreasing and E_21 is constant. As a last, I have shown that EF intensity has a minor effect on refractive index change.

Complete Hilbert-Space Ergodicity in Quantum Dynamics of Generalized Fibonacci Drives.

Ergodicity of quantum dynamics is often defined through statistical properties of energy eigenstates, as exemplified by Berry's conjecture in single-particle quantum chaos and the eigenstate thermalization hypothesis in many-body settings. In this work, we investigate whether quantum systems can exhibit a stronger form of ergodicity, wherein any time-evolved state uniformly visits the entire Hilbert space over time. We call such a phenomenon complete Hilbert-space ergodicity (CHSE), which is more akin to the intuitive notion of ergodicity as an inherently dynamical concept. CHSE cannot hold for time-independent or even time-periodic Hamiltonian dynamics, owing to the existence of (quasi)energy eigenstates which precludes exploration of the full Hilbert space. However, we find that there exists a family of aperiodic, yet deterministic drives with minimal symbolic complexity-generated by the Fibonacci word and its generalizations-for which CHSE can be proven to occur. Our results provide a basis for understanding thermalization in general time-dependent quantum systems.

Three-dimensional bound states of cylindrical quantum heterostructures with position-dependent mass carriers

We present a comprehensive spectral analysis of cylindrical quantum heterostructures by considering effective electronic carriers with position-dependent mass for five different kinetic-operator orderings. We obtain the bound energy eigenstates of particles in a three-dimensional cylindrical nanowire under a confining hyperbolic potential with both open and closed boundary conditions in the radial and the axial directions. In the present model we consider carriers with continuous mass distributions within the dot with abrupt mass discontinuities at the barriers, moving in a quantum dot that connects different substances. Continuity of mass and potential at the interfaces with the external layers result as a particular case. Our approach is mostly analytical and allows a precise comparison among von Roos ordering classes.

Highly accurate calculation of higher energy eigenvalues for the radial Schrödinger eigenproblems

In this paper, we derive an efficient numerical scheme for approximating the energy eigenvalues and corresponding wavefunctions of the radial Schrödinger eigenproblems defined on a semi-infinite domain for arbitrary values of the angular momentum number. The numerical scheme is based on the Chebyshev spectral collocation differentiation matrix method. In this scheme, at first, we redefine the radial Schrödinger eigenproblem on a finite interval by adopting an appropriate change of the independent variable. Then, by expanding the wavefunctions of the problem as a series of the Chebyshev polynomials of the first kind, as well as employing the differentiation matrices in order to determine the derivatives of Chebyshev polynomials at the collocation points, considered problem becomes a generalized eigenvalue problem. The convergence behavior and excellent performance of the proposed technique are illustrated through some numerical experiments of the most significant potentials in quantum mechanics. Compared with exact solutions (when available), and those reported previously in the literature, the current scheme achieves higher accuracy and efficiency even when high-index energy eigenvalues are computed.

Some properties of the eigenvalues of the sum geometric arithmetic means matrix of graphs

In this paper, we study some properties of the eigenvalues of sum geometric arithmetic energy of graphs and we obtain the values of some coefficients of the characteristic poly nominal of the sum geometric arithmetic means matrix by using some methods. The inverse of the sum geometric arithmetic means matrix of some graphs are also obtained.
 Keywords : The sum geometric arithmetic means matrix of a graph G. The characteristic polynomial of the sum geometric arithmetic means matrix of a graph G.

Thermodynamic state convertibility is determined by qubit cooling and heating

Thermodynamics plays an important role both in the foundations of physics and in technological applications. An operational perspective adopted in recent years is to formulate it as a quantum resource theory. At the core of this theory is the interconversion between athermality states, i.e. states out of thermal equilibrium. Here, we solve the question of how athermality can be used to heat and cool other quantum systems that are initially at thermal equilibrium. We then show that the convertibility between quasi-classical resources (resources that do not exhibit coherence between different energy eigenstates) is fully characterized by their ability to cool and heat qubits, i.e. by two of the most fundamental thermodynamical tasks on the simplest quantum systems.

Wavelet representation of hardcore bosons

We consider the one-dimensional Tonks–Girardeau gas with a space-dependent potential out of equilibrium. We derive the exact dynamics of the system when divided into n boxes and decomposed into energy eigenstates within each box. It is a representation of the wave function that is a mixture between real space and momentum space, with basis elements consisting of plane waves localized in a box, giving rise to the term ‘wavelet’. Using this representation, we derive the emergence of generalized hydrodynamics in appropriate limits without assuming local relaxation. We emphasize that a generalized hydrodynamic behaviour emerges in a high-momentum and short-time limit, in addition to the more common large-space and late-time limit, which is akin to a semi-classical expansion. In this limit, conserved charges do not require numerous particles to be described by generalized hydrodynamics. We also show that this wavelet representation provides an efficient numerical algorithm for a complete description of the out-of-equilibrium dynamics of hardcore bosons.

Tunneling for a semi-classical magnetic Schrödinger operator with symmetries

We are interested in decay estimates of the ground state (or the low energy eigenstates), outside the potential wells, for a semi-classical Magnetic Schrödinger operator with smooth coefficients PA (x, hD x ) = (hD x − µA(x))2 + V (x) on L 2(R d ). We shall essentially consider the case where µ is large. This kind of estimates, in case of Schrödinger operator without a magnetic field, have been studied by Agmon [1], also in the case of a Riemannian manifold M. Agmon estimates hold true for any h, but are particularly useful in the limit h → 0 when studying tunneling.

Sub-Diffraction Photon Trapping: The Possible Optical Energy Eigenstates within a Tiny Circular Aperture with a Finite Depth

One of the challenging riddles that is set by light is: do photons have wavefunctions like other elementary particles do? Wave–particle duality has been a prevailing fact since the beginning of quantum theory thought; in electromagnetism, light is already a kind of undulation, so what about the waves of probability then? Well, Quantum Field Theory (QFT) has a rigorous explanation and supports the idea when they are considered as fields of particles via second quantization; they do have wavefunctions of probability, and it does not have anything to do with the regular oscillations. They can be related to the energy and momentum signatures of harmonic oscillations, resembling an imitation of the behavior of a classical harmonic oscillator, which then has a wavefunction to solve the corresponding time-independent Schrödinger equation. For the last half century, electrical engineering has owned the best out of these implications of Quantum Electrodynamics (QED) and QFT by engineering better semiconductor techniques with finely miniaturized transistors and composite devices for digital electronics and optoelectronics fields. More importantly, these engineering applications have also greatly evolved into combined fields like quantum computing that have introduced a completely new and extraordinary world to electronics applications. The study takes advantage of the power of QFT to mathematically reveal the bosonic modes (Laguerre–Gaussian) that appear in a sub-diffraction cylindrical aperture. In this way, this may lead to the construction of the techniques and characteristics of room-temperature photonic quantum gates which can isolate photon modes under a diffraction limit.