The visualization of the space probability distribution for a particle moving in a double ring-shaped Coulomb potential

The analytical solutions to a double ring-shaped Coulomb potential (RSCP) are presented. The visualizations of the space probability distribution (SPD) are illustrated for the two-(contour) and three-dimensional (isosurface) cases. The quantum numbers (n, l, m) are mainly relevant for those quasi quantum numbers (n' ,l' ,m' ) via the double RSCP parameter c. The SPDs are of circular ring shape in spherical coordinates. The properties for the relative probability values (RPVs) P are also discussed. For example, when we consider the special case (n, l, m)=(6, 5, 0), the SPD moves towards two poles of axis z when the P increases. Finally, we discuss the different cases for the potential parameter b which is taken as negative and positive values for c>0 . Compared with the particular case b=0 , the SPDs are shrunk for b=-0.5 while spread out for b=0.5.Comment: 26 pages, 5 tables and 3 figures, Advances in High Energy Physics, in pres

The visualization of the space probability distribution for a moving particle I: in a single ring-shaped Coulomb potential

We first present the exact solutions of the single ring-shaped Coulomb potential and then realize the visualizations of the space probability distribution for a moving particle within the framework of this potential. We illustrate the two-(contour) and three-dimensional (isosurface) visualizations for those specifically given quantum numbers (n, l, m) essentially related to those so-called quasi quantum numbers (n',l',m') through changing the single ring-shaped Coulomb potential parameter b. We find that the space probability distributions (isosurface) of a moving particle for the special case and the usual case are spherical and circular ring-shaped, respectively by considering all variables in spherical coordinates. We also study the features of the relative probability values P of the space probability distributions. As an illustration, by studying the special case of the quantum numbers (n, l, m)=(6, 5, 1) we notice that the space probability distribution for a moving particle will move towards two poles of axis z as the relative probability value P increases. Moreover, we discuss the series expansion of the deformed spherical harmonics through the orthogonal and complete spherical harmonics and find that the principal component decreases gradually and other components will increase as the potential parameter b increases.Comment: 28 pages, 6 tables, 1 figure

Research on Wavelet Based Autofocus Evaluation in Micro-vision

AbstractThis paper presents the construction of two kinds of focusing measure operators defined in wavelet domain. One mechanism is that the Discrete Wavelet Transform (DWT) coefficients in high frequency subbands of in-focused image are higher than those of defocused one. The other mechanism is that the autocorrelation of an in-focused image filtered through Continuous Wavelet Transform (CWT) gives a sharper profile than blurred one does. Wavelet base, scaling factor and form to get the sum of high frequency energy are the key factors in constructing the operator. Two new focus measure operators are defined through the autofocusing experiments on the micro-vision system of the workcell for micro-alignment. The performances of two operators can be quantificationally evaluated through the comparison with two spatial domain operators Brenner Function (BF) and Squared Gradient Function (SGF). The focus resolution of the optimized DWT-based operators is 14% higher than that of BF and its computational cost is 52% approximately lower than BF's. The focus resolution of the optimized CWT-based operators is 41% lower than that of SGF whereas its computational cost is approximately 36% lower than SGF's. It shows that the wavelet based autofocus measure functions can be practically used in micro-vision applications

2-(2-Bromo­eth­yl)isoindoline-1,3-dione

The asymmetric unit of the title compound, C10H8BrNO2, contains three crystallographically independent mol­ecules. Two of the N—C—C—Br side chains adopt anti conformations [torsion angles = −179.8 (5) and −179.4 (4)°] and the other is gauche [−66.5 (6)°]. The crystal structure features short Br⋯O [3.162 (5) Å] contacts, C—H⋯O hydrogen bonds and numerous π–π stacking inter­actions [centroid–centroid separations = 3.517 (4)–3.950 (4) Å]

Diaqua­bis(benzyl­oxyacetato)copper(II)

In the title mononuclear complex, [Cu(C9H9O3)2(H2O)2], the CuII ion, located on an inversion center, is hexa­coordinated by four O atoms from two benzyl­oxyacetate ligands [Cu—O = 1.9420 (14) and 2.2922 (14) Å] and two water mol­ecules [Cu—O = 2.0157 (15) Å] in a distorted octa­hedral geometry. In the crystal structure, inter­molecular O—H⋯O hydrogen bonds link the mol­ecules into layers parallel to the bc plane

{2,2′-[1,1′-(Ethyl­enedioxy­dinitrilo)diethyl­idyne]di-1-naphtholato}copper(II)

The title complex, [Cu(C26H22N2O4)], is isostructural with its Ni analogue. All intramolecular distances and angles are very similar for the two structures, whereas the packing of the molecules, including C—H⋯O and C—H⋯π interactions, are slightly different