scholarly journals Quantum mechanics of electrons in crystal lattices

1931 ◽  
Vol 130 (814) ◽  
pp. 499-513 ◽  
Keyword(s):  
Three Dimensional ◽  
Matrix Elements ◽  
Periodic Field ◽  
Crystal Lattices ◽  
One Dimensional ◽  
Physical Problems ◽  
Radiative Transitions ◽  
First Approximation ◽  
The Matrix ◽  
The One

Introduction .–Through the work of Bloch our understanding of the behaviour of electrons in crystal lattices has been much advanced. The principal idea of Bloch’s theory is the assumption that the interaction of a given electron with the other particles of the lattice may be replaced in first approximation by a periodic field of potential. With this model an interpretation of the specific heat, the electrical and thermal conductivity, the magnetic susceptibility, the Hall effect, and the optical properties of metals could be obtained. The advantages and limitations inherent in the assumption of Bloch will be much the same as those encountered when replacing the interaction of the electrons in an atom by a suitable central shielding of the unclear field, as in the work of Thomas and Hartree. In the papers quoted a number of general results were given regarding the behaviour of electrons in any periodic field of potential. To obtain a clearer idea of the details of this behaviour with a view to the application in special problems, however, it appeared worth while to investigate the mechanics of electrons in periodic fields of potential somewhat similar to those met with in practice and of such nature that the energy values W and eigenfunctions Ψ of the wave-equation can actually be computed. It is the purpose of this article to discuss a case where the integration is possible. In Section 1 the energy values and in Section 2 the wave-functions in their dependence on the binding introduced by the potential field are discussed for the one dimensional problem. In Section 3 the matrix elements of the linear momentum, which furnish the electric current associated with the various stationary states, are well as the probability of radiative transitions between these states, are evaluated. In Section 4 the results are extended to the three dimensional case and those features considered which one may expect to find in the case of more general periodic fields of potential. Section 5 deals with some applications to physical problems.

scholarly journals Analysis of the Discrete Theory of Radiative Transfer in the Coupled “Ocean–Atmosphere” System: Current Status, Problems and Development Prospects

2020 ◽  
Vol 8 (3) ◽  
pp. 202 ◽  
Author(s):  
Viktor P. Afanas’ev ◽  
Alexander Yu. Basov ◽  
Vladimir P. Budak ◽  
Dmitry S. Efremenko ◽  
Alexander A. Kokhanovsky
Keyword(s):  
Radiative Transfer ◽  
Three Dimensional ◽  
Operator Method ◽  
Current Status ◽  
One Dimensional ◽  
Current State ◽  
The Matrix ◽  
The One ◽  
Discrete Theory ◽  
Development Prospects

In this paper, we analyze the current state of the discrete theory of radiative transfer. One-dimensional, three-dimensional and stochastic radiative transfer models are considered. It is shown that the discrete theory provides a unique solution to the one-dimensional radiative transfer equation. All approximate solution techniques based on the discrete ordinate formalism can be derived based on the synthetic iterations, the small-angle approximation, and the matrix operator method. The possible directions for the perspective development of radiative transfer are outlined.

On matrices whose eigenvalues are in arithmetic progression

1951 ◽  
Vol 47 (3) ◽  
pp. 585-590 ◽  
Author(s):  
P. T. Landsberg
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Arithmetic Progression ◽  
Matrix Elements ◽  
One Dimensional ◽  
The Matrix ◽  
The One ◽  
Matrix Problems ◽  
Image Position

The following matrix problems are well known in quantum mechanics:(a) The one-dimensional harmonic oscillator. Givendetermine the eigenvalues hjj of H, and the matrix elements of X, P if H is diagonal. It is found (Wigner (4)) that

Exploring the Multidimensional Facets of Authoritarianism: Authoritarian Aggression and Social Dominance Orientation

2008 ◽  
Vol 67 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Stefano Passini
Keyword(s):  
Social Dominance ◽  
Factor Model ◽  
Three Dimensional ◽  
Social Dominance Orientation ◽  
Model Fit ◽  
Regression Analyses ◽  
One Dimensional ◽  
Confirmatory Factor ◽  
The One ◽  
Better Than

The relation between authoritarianism and social dominance orientation was analyzed, with authoritarianism measured using a three-dimensional scale. The implicit multidimensional structure (authoritarian submission, conventionalism, authoritarian aggression) of Altemeyer’s (1981, 1988) conceptualization of authoritarianism is inconsistent with its one-dimensional methodological operationalization. The dimensionality of authoritarianism was investigated using confirmatory factor analysis in a sample of 713 university students. As hypothesized, the three-factor model fit the data significantly better than the one-factor model. Regression analyses revealed that only authoritarian aggression was related to social dominance orientation. That is, only intolerance of deviance was related to high social dominance, whereas submissiveness was not.

On the numerical study of thermohydrodynamics and biochemistry of inland water bodies

2021 ◽  
Author(s):  
Daria Gladskikh ◽  
Evgeny Mortikov ◽  
Victor Stepanenko
Keyword(s):  
Mixed Layer ◽  
Numerical Study ◽  
Three Dimensional ◽  
Water Bodies ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
One Dimensional ◽  
Lake Model ◽  
Upper Mixed Layer ◽  
The One

<p>The study of thermodynamic and biochemical processes of inland water objects using one- and three-dimensional RANS numerical models was carried out both for idealized water bodies and using measurements data. The need to take into account seiche oscillations to correctly reproduce the deepening of the upper mixed layer in one-dimensional (vertical) models is demonstrated. We considered the one-dimensional LAKE model [1] and the three-dimensional model [2, 3, 4] developed at the Research Computing Center of Moscow State University on the basis of a hydrodynamic code combining DNS/LES/RANS approaches for calculating geophysical turbulent flows. The three-dimensional model was supplemented by the equations for calculating biochemical substances by analogy with the one-dimensional biochemistry equations used in the LAKE model. The effect of mixing processes on the distribution of concentration of greenhouse gases, in particular, methane and oxygen, was studied.</p><p>The work was supported by grants of the RF President’s Grant for Young Scientists (MK-1867.2020.5, MD-1850.2020.5) and by the RFBR (19-05-00249, 20-05-00776). </p><p>1. Stepanenko V., Mammarella I., Ojala A., Miettinen H., Lykosov V., Timo V. LAKE 2.0: a model for temperature, methane, carbon dioxide and oxygen dynamics in lakes // Geoscientific Model Development. 2016. V. 9(5). P. 1977–2006.<br>2. Mortikov E.V., Glazunov A.V., Lykosov V.N. Numerical study of plane Couette flow: turbulence statistics and the structure of pressure-strain correlations // Russian Journal of Numerical Analysis and Mathematical Modelling. 2019. 34(2). P. 119-132.<br>3. Mortikov, E.V. Numerical simulation of the motion of an ice keel in stratified flow // Izv. Atmos. Ocean. Phys. 2016. V. 52. P. 108-115.<br>4. Gladskikh D.S., Stepanenko V.M., Mortikov E.V. On the influence of the horizontal dimensions of inland waters on the thickness of the upper mixed layer // Water Resourses. 2021.V. 45, 9 pages. (in press) </p>

scholarly journals The three dimensional stabilization of superconductivity at 12 K in the one dimensional organic superconductor (TMTSF)2PF6 under 11 kbar

1981 ◽  
Vol 42 (19) ◽  
pp. 445-449 ◽  
Author(s):  
A. Fournel ◽  
C. More ◽  
G. Roger ◽  
J.P. Sorbier ◽  
J.M. Delrieu ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Organic Superconductor ◽  
One Dimensional ◽  
The One

scholarly journals Hydraulic modeling of combined sewers overflow integrating the results of the SWMM and CFX models.

2021 ◽  
Vol 37 (1) ◽  
Author(s):  
D. Pulgarín ◽  
J. Plaza ◽  
J. Ruge ◽  
J. Rojas
Keyword(s):  
Three Dimensional ◽  
Hydraulic Modeling ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
One Dimensional ◽  
Ansys Cfx ◽  
Combined Sewer ◽  
The One ◽  
Swmm Model

This study proposes a methodology for the calibration of combined sewer overflow (CSO), incorporating the results of the three-dimensional ANSYS CFX model in the SWMM one-dimensional model. The procedure consists of constructing calibration curves in ANSYS CFX that relate the input flow to the CSO with the overflow, to then incorporate them into the SWMM model. The results obtained show that the behavior of the flow over the crest of the overflow weir varies in space and time. Therefore, the flow of entry to the CSO and the flow of excesses maintain a non-linear relationship, contrary to the results obtained in the one-dimensional model. However, the uncertainty associated with the idealization of flow methodologies in one dimension is reduced under the SWMM model with kinematic wave conditions and simulating CSO from curves obtained in ANSYS CFX. The result obtained facilitates the calibration of combined sewer networks for permanent or non-permanent flow conditions, by means of the construction of curves in a three-dimensional model, especially when the information collected in situ is limited.

One-Dimensional Harmonic Oscillator in Quantum Phase Space

2011 ◽  
Vol 110-116 ◽  
pp. 3750-3754
Author(s):  
Jun Lu ◽  
Xue Mei Wang ◽  
Ping Wu
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Quantum Phase ◽  
Space Representation ◽  
Wave Mechanics ◽  
One Dimensional ◽  
Matrix Mechanics ◽  
The Matrix ◽  
Quantum Wave ◽  
The One

Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.

Efficient computation of one-electron two-centre exchange integrals in ion–atom collisions

1976 ◽  
Vol 54 (9) ◽  
pp. 944-949 ◽  
Author(s):  
Alfred Msezane
Keyword(s):  
Principal Quantum Number ◽  
Computing Time ◽  
Translational Motion ◽  
Matrix Elements ◽  
Efficient Computation ◽  
Close Coupling ◽  
One Dimensional ◽  
State Approximation ◽  
The Matrix ◽  
Exchange Matrix

A scheme is presented for the reduction to one-dimensional integrals of any one-electron two-centre exchange matrix elements which incorporate the momentum associated with the translational motion of the electron. These elements are of the types occurring in close coupling-based treatments of ion–atom collisions. It is shown in a six state approximation, by coupling both eigenstates and pseudostates for the asymmetric He2+–H collision process, that computing time for the evaluation of the matrix elements is determined mainly by the number of different exponents in the matrix elements. The coupling of additional states with the same principal quantum number as the already coupled ones alters computing time insignificantly.

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