Stationary Schrodinger Equation Research Articles

Quantum mechanics denoising based channel estimation algorithm for mmWave massive MIMO systems

Channel estimation of millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) is crucial for the application of wireless transmission. The signal system is susceptible to external noise, which reduces the accuracy of channel estimation. The denoising of the received signal is a research hotspot and challenge for channel estimation. Therefore, this paper proposes a quantum mechanics denoising-based channel estimation method. The proposed quantum mechanics denoising-based algorithm has the advantages of not relying on the original conditions, no grid error, and strong adaptive ability. The first part is that the received noisy signal is denoised. The conversion between the signal model of mmWave massive MIMO and the physical model of quantum mechanics needs to be solved. The received noisy signal is equivalent to the potential of the stationary Schrodinger equation. Then, the Hamiltonian matrix is constructed by the received signal and the corresponding eigenvalues and eigenvectors are calculated. The eigenvectors of the Hamiltonian matrix are related to the energy of Schrodinger equation, which are determined to the adaptive basis. In addition, the received signal is projected onto the adaptive basis to calculate the coefficients. The denoised received signal is reconstructed through the soft threshold processing of the coefficients. The second part is that channel estimation is performed on the denoised received signal using the l1/2-singular value decomposition (SVD)-based algorithm. The angle parameters are iteratively moved to the actual values by gradient descent. Besides, the initial values of the angles are obtained through the SVD preprocessing method. Simulation results show that the proposed quantum mechanics denoising-based method exhibits good estimation accuracy.

Simultaneous effect study of eccentricity and capping matrix on effective dielectric function in spheroidal CdSe/ZnSe core/shell quantum dot

We have investigated, in this paper, the simultaneous effect of eccentricity and encapsulating matrix on the effective dielectric function εeff in spheroid-shaped CdSe/ZnSe core/shell quantum dot. The calculations of electronic energy levels and the corresponding wavefunctions were carried out by solving the stationary Schrodinger equation assuming the Effective Mass Approximation (EMA) and taking in consideration the Compact Density Matrix (CDM). The investigated effective dielectric constant εeff is very important due to its relevance in the manipulation of the optoelectronic properties as required in the design of new sophisticated devices. It turned out that the embedding matrix has a great effect on the resonance peaks of εeff. Also, the eccentricity, key parameter in spheroid-shaped core/shell structure, play a fundamental role in the manipulation of the energy levels, transition energies as well as the effective dielectric function resonance peaks.

Quantum-mechanical model of dielectric losses in nanometer layers of solid dielectrics with hydrogen bonds at ultra-low temperatures

Upon based the finite difference methods construct the solutions for Liouville quantum kinetic equation linearized by the external field, in complex with the stationary Schrodinger equation and the Poisson operator equation, for an ensemble of non-interacting hydrogen ions (protons) migrating in the field of a crystal lattice perturbed by a variable polarizing field. The influence of the phonon subsystem is not taken into account. The equilibrium (non-balanced) proton density matrix is calculated using quantum Boltzmann statistics. The temperature spectra of dielectric losses tangent angle for hydrogen bonded crystals (HBC) in a wide temperature range (50–550 K) are calculated. At the theoretical level detected the effects of nano-crystalline states (1–10 nm) during the polarization of HBC in the region of ultra-low temperatures (4–25 K).

Bounded Solutions of the Stationary Schrodinger Equation with Finite Energy Integral on Model Manifolds

Bounded Solutions of the Stationary Schrodinger Equation with Finite Energy Integral on Model Manifolds

A Sharp Version of Phragmén–Lindelöf Type Theorem for the Stationary Schrödinger Equation

In this paper, we use Cauchy’s integral formula with respect to the Schrodinger operator to investigate the asymptotic behavior for the stationary Schrodinger equation when it is applied to some Schrodinger integral equations. Moreover, from using the theoretical significance of the result obtained, we draw conclusions about the Phragmen–Lindelof principle of weak solutions of the equations. Furthermore, a sharp version of this principle is also given.

On One Implementation of the Numerov Method for the One-Dimensional Stationary Schrödinger Equation

We present accurate numerical results for the one-dimensional stationary Schrodinger equation in the case of three quantum problems: quantum harmonic oscillator, radial Schrodinger equation for a Hydrogen atom, and a particle penetration through the potential barrier. All of them were solved by the Numerov method with high accuracy and we plot their wave functions using the results of numerical calculations. Furthermore, we offer accurate numerical methods for solving boundary value problems, eigenvalue problems, matrix elimination.

Comments on the paper “Exact and Poisson summation thermodynamic properties for diatomic molecules with shifted Tietz potential”

Ikot et al. (Indian J Phys 93:1171, 2019) showed an analytical solution of the stationary Schrodinger equation for cesium dimers $$Cs_2$$ at the electronic state $$(3^3\Sigma _g^+)$$ with the shifted Tietz–Wei potential. We point out, however, a miscalculation in solving this equation which had led to erroneous conclusions, particularly about the energy eigenvalues, the partition function and all derived thermodynamic properties. By the way, we present the right expression of the rotational–vibrational energy which shows an excellent agreement with the experimental data.

On the Use of B-Splines as Ritz Variational Basis Functions to Solve the Schrodinger Equation (TISE) for a constrained free Quantum Particle

B-Splines as piecewise adaptation of Bernstein polynomials (aka, B-polys) are widely used as Ritz variational basis functions in solving many problems in the fields of quantum mechanics and atomic physics. In this paper they are used to solve the 1-D stationary Schrodinger equation (TISE) for a free quantum particle subject to a fixed domain length by using the Python software SPLIPY with different sets of computation parameters. In every case it was found that over 60 percent of energy levels had excellent accuracy thereby proving that the use of B-spline collocation is a preferred method.

Kink and Anti-Kink Soliton for Stationary Cubic-Quintic Discrete Nonlinear Schrodinger Equation

Kink and Anti-Kink Soliton for Stationary Cubic-Quintic Discrete Nonlinear Schrodinger Equation

Effect of structural properties of InAs/GaAs QDs on optoelectronic devices

The efficiency of Quantum Dots (QDs) devices, is strongly related to their distribution, density and size, and shape in a QD array.The small size of the QD structure leads to a three-dimensional confinement of the electronic energies. The energy levels of electrons in QDs depends on their size – the smaller the quantum dot, the greater the amount of energy required to give electrons transitions to the next level. In this paper, we report the simulation results of pyramidal InAs/GaAs QDs coupled to Wetting Layer (WL). To investigate the effects of the InAs/GaAs QDs size on the electronic states and the transition energy. In the first part, we calculate the strain distribution to find the band edges, while the second one concerns the variation of the electronic transition energy as a function of quantum dots size by solving the stationary Schrodinger equation. This work shows clearly that the structural properties of QDs have strong effects on optoelectronic properties devices based on these nanostructures.

Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials

<p style='text-indent:20px;'>In this paper, we study the one-dimensional stationary Schrödinger equation with quasi-periodic potential <inline-formula><tex-math id="M1">\begin{document}$ u(\omega t) $\end{document}</tex-math></inline-formula>. We show that if the frequency vector <inline-formula><tex-math id="M2">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> is sufficient large, the Schrödinger equation admits two linear independent Floquet solutions for a set of positive measure of energy <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula>. In contrast with previous results, the conditions of small potential <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> or large energy <inline-formula><tex-math id="M5">\begin{document}$ E $\end{document}</tex-math></inline-formula> are no longer needed.

Quantum tunneling of molecules through compound barriers

The use of hydrogen and helium as a fuel requires well-developed technologies for the separation of gas mixtures, in particular, separation of isotopes of an element. The presence of quantum effects makes such processes possible, and the proposed system of two potential barriers enhances the differences in the passage of isotopes through two layers of matter providing high separation efficiency. The probability of atoms passing through composite barriers is derived from the solving of stationary Schrodinger equation.

Dimensions of Solution Spaces of the Schrodinger Equation with Finite Dirichlet Integral on Non-compact Riemannian Manifolds

Exact estimations of dimensions of spaces of bounded solutions of stationary Schrodinger equation with finite Dirichlet integral in terms of massive sets are obtained. It is proved that dimension of spaces of bounded solutions of this equation is not less than number of disjoint qD-massive subsets of manifold. This paper partly extends, the results of A.A. Grigor’yan, A.G. Losev (2017).

Monotonicity and local uniqueness for the Helmholtz equation

This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schrodinger) equation (Δ+k2q)u=0 in a bounded domain for fixed nonresonance frequency k>0 and real-valued scattering coefficient function q. We show a monotonicity relation between the scattering coefficient q and the local Neumann-to-Dirichlet operator that holds up to finitely many eigenvalues. Combining this with the method of localized potentials, or Runge approximation, adapted to the case where finitely many constraints are present, we derive a constructive monotonicity-based characterization of scatterers from partial boundary data. We also obtain the local uniqueness result that two coefficient functions q1 and q2 can be distinguished by partial boundary data if there is a neighborhood of the boundary part where q1≥q2 and q1≢q2.

Absorption spectra of different InGaN-GaN QD-shapes coupled to wetting layer

Wetting layer-dependent optical absorption spectra of wurtzite (In,Ga)N/GaN QDs corresponding to different shapes are investigated in this paper. Based on the numerical finite element method, the stationary Schrodinger equation is resolved. Using one-band parabolic effective mass, electronic states in the conduction band are computed under the finite potential barrier. Strained confining potential and band gap values were utilized. Strong wetting layer thickness and size impacts on P-to-S transition energy, dipole moment, oscillator strength and optical absorption spectra were illustrated. Significant red shift is induced as the WL thickness increases. Two behaviors, blue and red shifts, have obtained limited by a critical dot size.

Development of the perturbation theory using polynomial solutions

The number of quantum systems for which the stationary Schrodinger equation is exactly solvable is very limited. These systems constitute the basic elements of the quantum theory of perturbation. The exact polynomial solutions for real quantum potential systems provided by the use of Lagrange interpolation allows further development of the quantum perturbation theory. In fact, the first order of correction for the value of the energy appears to be sufficient since the chosen perturbation Hamiltonian is very small or even negligible compared to the main Hamiltonian. Here, we use the perturbation theory to derive polynomial solutions, and we then find that our approximated results agree very well with previous published or numerically achieved ones. We believe that this study is an operational tool for the verification and improvement of numerical and approximate methods.

A Schrödinger Potential Conditionally Integrable in Terms of the Hermite Functions

We introduce a biconfluent Heun potential well for the one-dimensional stationary Schrodinger equation which is composed of a confining fraction-power term and a repulsive centrifugal-barrier core. This is a conditionally integrable potential in that the strength of the centrifugal barrier is fixed to a constant. The potential supports a countable infinite number of bound states. We present the general solution of the Schrodinger equation, deduce the exact equation for the energy spectrumand derive a highly accurate approximation for energy levels. The bound state wave functions are written as irreducible linear combinations with constant coefficients of two Hermite functions of a scaled and shifted argument.

Bubbling above the threshold of the scalar curvature in dimensions four and five

On any closed manifold $$(M^n,g)$$ of dimension $$n\in \{4,5\}$$ we exhibit new blow-up configurations for perturbations of a purely critical stationary Schrodinger equation. We construct positive solutions which blow-up as the sum of two isolated bubbles, one of which concentrates at a point $$\xi $$ where the potential k of the equation satisfies $$\begin{aligned} k(\xi ) > \frac{n-2}{4(n-1)} S_g(\xi ), \end{aligned}$$ where $$S_g$$ is the scalar curvature of $$(M^n,g)$$ . The latter condition requires the bubbles to blow-up at different speeds and forces us to work at an elevated precision. We take care of this by performing a construction which combines a priori asymptotic analysis methods with a Lyapounov–Schmidt reduction.

A note on the use of analytical and domain discretisation methods for the analysis of some phenomena in engineering physics

The paper reviews some applications of the well-established analytical and domain discretisation methods (finite difference method-FDM, finite element method-FEM) in the modelling of the magnetohydrodynamics (MHD) phenomena for the fusion-related research and quantum structures for use in nanodevices. Hence, Grad-Shafranov Equation (GSE) for the plasma equilibrium has been implemented for certain simple excitation forms, and the results obtained for the rectangular plasma have been presented. Furthermore, the stationary Schrodinger equation is solved analytically and numerically via FDM and FEM, respectively.

The Extension of the Monte Carlo Method for Neutron Transfer Problems Calculating to the Problems of Quantum Mechanics

There are several methods of numerical solution of eigenvalue problems by the Monte Carlo method, which are used in the calculation of nuclear reactors. This paper is devoted to the investigation of the possibility of using such methods for solving the stationary Schrodinger equation. The latter equation can easily be transformed into the form of an integral equation of the first kind, very similar to those integral equations that arise in problems of nuclear power. The Monte Carlo method for this form of the stationary Schrodinger equation looks very attractive, since it naturally parallels and is very convenient for calculations on multiprocessor systems. In addition, in this case it is necessary to operate with functions defined on a large-dimensional space. This is also natural for the Monte Carlo method. It is described how the methods long used for the calculation of nuclear reactors are transformed for this case. The main problem is that the wave function of fermions changes its sign under a permutation of identical particles, and can not be nonnegative. The proposed approach is significantly different from the known methods of applying the Monte Carlo method to quantum mechanical problems. In this paper, several examples of the successful application of the proposed new method are given.