The Solution of the Relativistic Schrodinger Equation for the δ′\delta'-Function Potential in 1-dimension Using Cutoff Regularization

We study the relativistic version of Schr\"odinger equation for a point particle in 1-d with potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states. The initial calculations show that the reciprocal of the bare coupling constant is ultra-violet divergent, and the resultant expression cannot be renormalized in the usual sense. Therefore a general procedure has been developed to derive different physical properties of the system. The procedure is used first on the non-relativistic case for the purpose of clarification and comparisons. The results from the relativistic case show that this system behaves exactly like the delta function potential, which means it also shares the same features with quantum field theories, like being asymptotically free, and in the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point.Comment: 32 pages, 5 figure

Fate of Accidental Symmetries of the Relativistic Hydrogen Atom in a Spherical Cavity

The non-relativistic hydrogen atom enjoys an accidental $SO(4)$ symmetry, that enlarges the rotational $SO(3)$ symmetry, by extending the angular momentum algebra with the Runge-Lenz vector. In the relativistic hydrogen atom the accidental symmetry is partially lifted. Due to the Johnson-Lippmann operator, which commutes with the Dirac Hamiltonian, some degeneracy remains. When the non-relativistic hydrogen atom is put in a spherical cavity of radius $R$ with perfectly reflecting Robin boundary conditions, characterized by a self-adjoint extension parameter $\gamma$, in general the accidental $SO(4)$ symmetry is lifted. However, for $R = (l+1)(l+2) a$ (where $a$ is the Bohr radius and $l$ is the orbital angular momentum) some degeneracy remains when $\gamma = \infty$ or $\gamma = \frac{2}{R}$. In the relativistic case, we consider the most general spherically and parity invariant boundary condition, which is characterized by a self-adjoint extension parameter. In this case, the remnant accidental symmetry is always lifted in a finite volume. We also investigate the accidental symmetry in the context of the Pauli equation, which sheds light on the proper non-relativistic treatment including spin. In that case, again some degeneracy remains for specific values of $R$ and $\gamma$.Comment: 27 pages, 7 figure

Majorana Fermions in a Box

Majorana fermion dynamics may arise at the edge of Kitaev wires or superconductors. Alternatively, it can be engineered by using trapped ions or ultracold atoms in an optical lattice as quantum simulators. This motivates the theoretical study of Majorana fermions confined to a finite volume, whose boundary conditions are characterized by self-adjoint extension parameters. While the boundary conditions for Dirac fermions in $(1+1)$-d are characterized by a 1-parameter family, $\lambda = - \lambda^*$, of self-adjoint extensions, for Majorana fermions $\lambda$ is restricted to $\pm i$. Based on this result, we compute the frequency spectrum of Majorana fermions confined to a 1-d interval. The boundary conditions for Dirac fermions confined to a 3-d region of space are characterized by a 4-parameter family of self-adjoint extensions, which is reduced to two distinct 1-parameter families for Majorana fermions. We also consider the problems related to the quantum mechanical interpretation of the Majorana equation as a single-particle equation. Furthermore, the equation is related to a relativistic Schr\"odinger equation that does not suffer from these problems.Comment: 23 pages, 2 figure

Asymptotic Freedom, Dimensional Transmutation, and an Infra-red Conformal Fixed Point for the δ\delta-Function Potential in 1-dimensional Relativistic Quantum Mechanics

We consider the Schr\"odinger equation for a relativistic point particle in an external 1-dimensional $\delta$-function potential. Using dimensional regularization, we investigate both bound and scattering states, and we obtain results that are consistent with the abstract mathematical theory of self-adjoint extensions of the pseudo-differential operator $H = \sqrt{p^2 + m^2}$. Interestingly, this relatively simple system is asymptotically free. In the massless limit, it undergoes dimensional transmutation and it possesses an infra-red conformal fixed point. Thus it can be used to illustrate non-trivial concepts of quantum field theory in the simpler framework of relativistic quantum mechanics

Modeling and Optimization of Phosphate Recovery from Industrial Wastewater and Precipitation of Solid Fertilizer using Experimental Design Methodology

In this work, the experimental design methodology is applied to optimize phosphate salts precipitation as struvite and others applied in soil fertilization from treated industrial wastewater stream. This is a process to maximize phosphate recovery percentage from inlet wastewater stream containing interfering foreign ions. Therefore, these optimized conditions could be used as input data for engineering design-software for successive equipment required in wastewater treatment plant. A four factors Box–Behnken experimental design was used to model and optimize the operating parameters. The optimum operating conditions were quite efficient in trapping 86.10 % recovered phosphates in industrial stream, and 92.6 % in synthetic solution at pH of 10.89, time of reaction of 34.76 min, temperature of 25.23 °C and R of 2.25 with an insignificance effect for molar ratio (R) between Mg and PO4 ions. If these optimal parameters were shifted, the reached recovery percentage would decrease with the precipitated struvite. The precipitated salts were subjected to characterization through different chemical techniques confirming the presence of struvite with schertelite as a mixed slow release fertilizer

Study on Quality of Pair Distribution Function for Direct Space Approach of Structure Investigation

Study of the structure characteristics of solid materials is a key for development of technological applications. Potential of direct space approach for structure determination and refinement using powder X-ray diffraction data depend on the quality of pair distribution function (PDF) plot. So, the effect of data collection conditions and diffractogram characteristics on the quality of PDF plot has been investigated in detail. In addition, errors and possible tolerance have been estimated. Some parameters affect only either the X-ray diffractogram or PDF plots and others affect both. Considering the errors and tolerance, direct space approach can be confidently used for structure refinement, where the error did not exceed 10.0 % for inter-atomic radial distance longer than » 2.0 ? and 5.0 % for longer than » 4.0 ?, which is accepted for structure refinement. As tolerance is considered, every time the value of the lattice parameter is changed to smaller or larger than the correct value (+ 8.0 %), it comes back to the initial correct one. Although, advanced synchrotron radiation shows better data, conventional source can be used successfully for structure investigation applying direct space approach

Precise Cerebrovascular Segmentation

© 2020 IEEE. Analyzing cerebrovascular changes using Time-of-Flight Magnetic Resonance Angiography (ToF-MRA) images can detect the presence of serious diseases and track their progress, e.g., hypertension. Such analysis requires accurate segmentation of the vasculature from the surroundings, which motivated us to propose a fully automated cerebral vasculature segmentation approach based on extracting both prior and current appearance features that capture the appearance of macro and micro-vessels. The appearance prior is modeled with a novel translation and rotation invariant Markov-Gibbs Random Field (MGRF) of voxel intensities with pairwise interaction analytically identified from a set of training data sets, while the current appearance is represented with a marginal probability distribution of voxel intensities by using a Linear Combination of Discrete Gaussians (LCDG) whose parameters are estimated by a modified Expectation-Maximization (EM) algorithm. The proposed approach was validated on 190 data sets using three metrics, which revealed high accuracy compared to existing approaches

Weak mutually unbiased bases

Quantum systems with variables in ${\mathbb Z}(d)$ are considered. The properties of lines in the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space of these systems, are studied. Weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases, is equal to $d^{-1/2}$ or alternatively to one of the $d_i^{-1/2},0$ (where $d_i$ is a divisor of $d$ apart from $d,1$). They are designed for the geometry of the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space, in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin. In the special case of prime $d$, there are no divisors of $d$ apart from $1,d$ and the weak mutually unbiased bases are mutually unbiased bases

Peer Social Acceptance of Students with Special Needs

The purpose of this research is to investigate how kids with ASD are accepted socially by their peers. Peer social acceptability may be seen in the manner in which autistic kids are treated by their peers as well as in how they show and demonstrate their desire to participate in various activities. In addition, the many kinds of connections that may be seen between ordinary students and pupils who have ASD. This research is a qualitative study that focuses on description. The participants in this research were classmates and their respective professors. Interviews and observations were the methods of data collection that were employed for this investigation. The method of data analysis that was employed was called descriptive analysis, and it included reducing the amount of data, presenting the data, and deriving conclusions from the data. Verifying the accuracy of the authors claims by using methods such as triangulation, extended observations, and consultation with others. Students who have ASD spectrum disorder may participate in social activities with their peers. The classroom instructors support and understanding helps ordinary students better appreciate the condition of students with ASD. The teacher also understands children with ASD when they have tantrums and may be an aid when students with ASD are having problems. Social interactions and group relations are the types of relationships that may develop between children with ASD and their typical classmates. Regular students are able to create group interactions with students who have ASD with the assistance of the class teachers promotion of the formation of study groups