Numerical Methods for the Evaluation of the Löwdin α-Function

2005 ◽  
Vol 70 (8) ◽  
pp. 1035-1054 ◽  
Author(s):  
Nemanja Sovic ◽  
James D. Talman
Keyword(s):  
Wave Function ◽  
Angular Momentum ◽  
Numerical Methods ◽  
Rate Of Convergence ◽  
Wave Functions ◽  
Analytic Methods ◽  
Nuclear Attraction ◽  
Numerical Approaches ◽  
Löwdin Α Function

The problem of expanding an angular momentum wave function centered at one point in terms of angular momentum wave functions centered at another point is analysed. The emphasis is on obtaining methods that can be applied to functions that are defined numerically, in contrast to analytic methods. Three numerical approaches are described, and it is found that one leads to extremely accurate results. The question of the rate of convergence of the resulting series is discussed, and results of the application of the expansion to the calculation of nuclear attraction three-center integrals, and electron-electron four-center integrals are presented.

Formation of an alpha-group in the shell-model of a heavy nucleus

1957 ◽  
Vol 53 (2) ◽  
pp. 494-507
Author(s):  
Charles Strachan
Keyword(s):  
Wave Function ◽  
Angular Momentum ◽  
Shell Model ◽  
Relative Weight ◽  
Wave Functions ◽  
Variation Principle ◽  
Nuclear Wave Function ◽  
Spin Orbit Coupling ◽  
Spin Orbit ◽  
Particle Group

ABSTRACTBy means of a variation principle applied to a nuclear wave function for four nucleons in the field of a closed core, equations are derived giving the relative proportions of a component describing an alpha-particle group and components describing the four nucleons in shell-model states with spin-orbit coupling wave functions, the two neutron states and the two proton states each paired to give zero resultant angular momentum. Numerical calculations, taking into account only 1h2/3 states for protons and the 2g3/2 states for neutrons, give the alpha-group wave function a relative weight (in the linear combination) of about 1%.

Low-lying collective bands in the even tungsten isotopes 180–186W

1989 ◽  
Vol 67 (5) ◽  
pp. 479-484 ◽  
Author(s):  
R. Sahu
Keyword(s):  
Wave Function ◽  
Angular Momentum ◽  
Quadrupole Interaction ◽  
Asymmetry Parameter ◽  
Degrees Of Freedom ◽  
Energy Levels ◽  
Wave Functions ◽  
Electromagnetic Properties ◽  
Matrix Elements ◽  
Rigid Model

The γ-rigid model has been used to study the energy levels and the electromagnetic properties of 180, 182, 184, 186W. In this model, the intrinsic wave functions are obtained using the pairing plus the quadrupole–quadrupole interaction Hamiltonian of Baranger and Kumar. Good angular momentum states are projected approximately from such a triaxially symmetric intrinsic wave function. This model assumes the nucleus to be γ rigid but soft in the β degrees of freedom. The asymmetry parameter γ for a given nucleus is extracted using the experimental energies of the first 2+ and second 2+ states within the framework of the Davydov–Filippov model. The symmetry parameter β for each J state is determined from the minimization of the projected energy. The calculated energy levels of the ground and the 7 band, the B(E2) values, the electromagnetic moments, the E2 and E4 matrix elements, and the B(E2) ratios agree quite well with experimental results.

Gravity as the curvature of the wave function of the universe.

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Wave Function ◽  
Wave Functions ◽  
Main Task ◽  
Reference Systems ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Principle Of Equivalence ◽  
Relative Wave Function ◽  
New Equation ◽  
The Universe

Modern general theory of relativity considers gravity as the curvature of space-time. The theory is based on the principle of equivalence. All bodies fall with the same acceleration in the gravitational field, which is equivalent to locally accelerated reference systems. In this article, we will affirm the concept of gravity as the curvature of the relative wave function of the Universe. That is, a change in the phase of the universal wave function of the Universe near a massive body leads to a change in all other wave functions of bodies. The main task is to find the form of the relative wave function of the Universe, as well as a new equation of gravity for connecting the curvature of the wave function and the density of matter.

scholarly journals Feasibility Study on Oil Droplet Flow Investigations Inside Aero Engine Bearing Chambers: PDPA Techniques in Combination With Numerical Approaches

1995 ◽  
Author(s):  
A. Glahn ◽  
M. Kurreck ◽  
M. Willmann ◽  
S. Wittig
Keyword(s):  
Numerical Methods ◽  
Flow Field ◽  
Flow Analysis ◽  
Field Analysis ◽  
Oil Droplet ◽  
Droplet Flow ◽  
Aero Engine ◽  
Secondary Air ◽  
Numerical Approaches ◽  
Engine Bearing

The present paper deals with oil droplet now phenomena in aero engine bearing chambers. An experimental investigation of droplet sizes and velocities utilizing a Phase Doppler Particle Analyzer (PDPA) has been performed for the first time in bearing chamber atmospheres under real engine conditions. Influences of high rotational speeds are discussed for individual droplet size classes. Although this is an important contribution to a better understanding of the droplet flow impact on secondary air/oil system performance, an analysis of the droplet flow behaviour requires an incorporation of numerical methods because detailed measurements as performed here suffer from both strong spatial limitations with respect to the optical accessibility in real engine applications and constraints due to the extremely time consuming nature of an experimental flow field analysis. Therefore, further analysis is based on numerical methods. Droplets characterized within the experiments are exposed to the flow field of the gaseous phase predicted by use of our well-known CFD code EPOS. The droplet trajectories and velocities are calculated within a Lagrangian frame of reference by forward numerical integration of the particle momentum equation. This paper has been initiated rather to show a successful method of bearing chamber droplet flow analysis by a combination of droplet sizing techniques and numerical approaches than to present field values as a function of all operating parameters. However, a first insight into the complex droplet flow phenomena is given and specific problems in bearing chamber heat transfer are related to the droplet flow.

scholarly journals INTEGRAL TRANSFORM TECHNIQUE FOR MESON WAVE FUNCTIONS

1996 ◽  
Vol 11 (20) ◽  
pp. 1611-1626 ◽  
Author(s):  
A.P. BAKULEV ◽  
S.V. MIKHAILOV
Keyword(s):  
Wave Function ◽  
Sum Rules ◽  
Integral Transform ◽  
Wave Functions ◽  
Sum Rule ◽  
New Approach ◽  
Exactly Solvable ◽  
The Stability ◽  
The Masses ◽  
Integral Transform Technique

In a recent paper1 we have proposed a new approach for extracting the wave function of the π-meson φπ(x) and the masses and wave functions of its first resonances from the new QCD sum rules for nondiagonal correlators obtained in Ref. 2. Here, we test our approach using an exactly solvable toy model as illustration. We demonstrate the validity of the method and suggest a pure algebraic procedure for extracting the masses and wave functions relating to the case under investigation. We also explore the stability of the procedure under perturbations of the theoretical part of the sum rule. In application to the pion case, this results not only in the mass and wave function of the first resonance (π′), but also in the estimation of π″-mass.

The Wave Theory of the Electron

1928 ◽  
Vol 24 (4) ◽  
pp. 501-505 ◽  
Author(s):  
J. M. Whittaker
Keyword(s):  
Wave Function ◽  
Differential Equations ◽  
Fourth Order ◽  
Commutative Algebra ◽  
Wave Theory ◽  
Wave Functions ◽  
Linear Differential Equations ◽  
Wave Mechanics ◽  
First Order ◽  
Linear Differential

In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.

LONG RANGE FORCES BETWEEN HYDROGEN MOLECULES

1955 ◽  
Vol 33 (11) ◽  
pp. 668-678 ◽  
Author(s):  
F. R. Britton ◽  
D. T. W. Bean
Keyword(s):  
Wave Function ◽  
Long Range ◽  
Ground State ◽  
Close Agreement ◽  
Van Der Waals ◽  
Wave Functions ◽  
Numerical Factor ◽  
Hydrogen Molecules ◽  
Static Quadrupole

Long range forces between two hydrogen molecules are calculated by using methods developed by Massey and Buckingham. Several terms omitted by them and a corrected numerical factor greatly change results for the van der Waals energy but do not affect their results for the static quadrupole–quadrupole energy. By using seven approximate ground state H2 wave functions information is obtained regarding the dependence of the van der Waals energy on the choice of wave function. The value of this energy averaged over all orientations of the molecular axes is found to be approximately −11.0 R−6 atomic units, a result in close agreement with semiempirical values.

scholarly journals DISTRIBUTION OF MAXIMA OF THE ANTISYMMETRIZED WAVE FUNCTION FOR THE NUCLEONS OF A CLOSED-SHELL AND FOR THE NUCLEONS OF ALL CLOSED-SHELLS IN A NUCLEUS.

2020 ◽  
Vol 15 ◽  
pp. 57
Author(s):  
G. S. Anagnostatos
Keyword(s):  
Wave Function ◽  
Mathematical Problem ◽  
Closed Shell ◽  
Wave Functions ◽  
Pictorial Representation ◽  
One Dimensional ◽  
Regular Polyhedra ◽  
Closed Shell Nuclei ◽  
Simple Systems ◽  
Closed Shells

The significant features of exchange symmetry are displayed by simple systems such as two identical, spinless fermions in a one-dimensional well with infinite walls. The conclusion is that the maxima of probability of the antisymmetrized wave function of these two fermions lie at the same positions as if a repulsive force (of unknown nature) was applied between these two fermions. This conclusion is combined with the solution of a mathematical problem dealing with the equilibrium of identical repulsive particles (of one or two kinds) on one or more spheres like neutrons and protons on nuclear shells. Such particles are at equilibrium only for specific numbers of particles and, in addition, if these particles lie on the vertices of regular polyhedra or their derivative polyhedra. Finally, this result leads to a pictorial representation of the structure of all closed shell nuclei. This representation could be used as a laboratory for determining nuclear properties and corresponding wave functions.

Spinless relativistic particle in energy-dependent potential and normalization of the wave function

Open Physics ◽  
2014 ◽  
Vol 12 (6) ◽  
Author(s):  
Amar Benchikha ◽  
Lyazid Chetouani
Keyword(s):  
Wave Function ◽  
Spectral Decomposition ◽  
Relativistic Particle ◽  
Wave Functions ◽  
Integral Approach ◽  
Normalizing Constant ◽  
Klein Gordon ◽  
Dependent Potential ◽  
Energy Dependent ◽  
Coulomb Potentials

AbstractThe problem of normalization related to a Klein-Gordon particle subjected to vector plus scalar energy-dependent potentials is clarified in the context of the path integral approach. In addition the correction relating to the normalizing constant of wave functions is exactly determined. As examples, the energy dependent linear and Coulomb potentials are considered. The wave functions obtained via spectral decomposition, were found exactly normalized.

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