For centuries knowledge meant proven knowledge — proven either by the power of the intellect or by the evidence of the senses. Wisdom and intellectual integrity demanded that one must desist from unproven utterances and minimize, even in thought, the gap between speculation and established knowledge. The proving power of the intellect or the senses was questioned by the sceptics more than two thousand years ago; but they were browbeaten into confusion by the glory of Newtonian physics. Einstein’s results again turned the tables and now very few philosophers or scientists still think that scientific knowledge is, or can be, proven knowledge. But few realize that with this the whole classical structure of intellectual values falls in ruins and has to be replaced: one cannot simply water down the ideal of proven truth - as some logical empiricists do — to the ideal of’probable truth’1 or — as some sociologists of knowledge do — to ‘truth by [changing] consensus’.2

Falsification and the Methodology of Scientific Research Programmes

In der vorliegenden Arbeit werden zunachst exakte Definitionen der Worte: Ort, Geschwindigkeit, Energie usw. (z. B. des Elektrons) aufgestellt, die auch in der Quantenmechanik Gultigkeit behalten, und es wird gezeigt, das kanonisch konjugierte Grosen simultan nur mit einer charakteristischen Ungenauigkeit bestimmt werden konnen (§ 1). Diese Ungenauigkeit ist der eigentliche Grund fur das Auftreten statistischer Zusammenhange in der Quantenmechanik. Ihre mathematische Formulierung gelingt mittels der Dirac-Jordanschen Theorie (§ 2). Von den so gewonnenen Grundsatzen ausgehend wird gezeigt, wie die makroskopischen Vorgange aus der Quantenmechanik heraus verstanden werden konnen (§ 3). Zur Erlauterung der Theorie werden einige besondere Gedankenexperimente diskutiert (§ 4).

Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik

Preface Acknowledgements Notation Part I. Overview and Background Topics: 1. Introduction 2. Overview 3. Theoretical background 4. Periodic solids and electron bands 5. Uniform electron gas and simple metals Part II. Density Functional Theory: 6. Density functional theory: foundations 7. The Kohn-Sham ansatz 8. Functionals for exchange and correlation 9. Solving the Kohn-Sham equations Part III. Important Preliminaries on Atoms: 10. Electronic structure of atoms 11. Pseudopotentials Part IV. Determination of Electronic Structure, The Three Basic Methods: 12. Plane waves and grids: basics 13. Plane waves and grids: full calculations 14. Localized orbitals: tight binding 15. Localized orbitals: full calculations 16. Augmented functions: APW, KKR, MTO 17. Augmented functions: linear methods Part V. Predicting Properties of Matter from Electronic Structure - Recent Developments: 18. Quantum molecular dynamics (QMD) 19. Response functions: photons, magnons ... 20. Excitation spectra and optical properties 21. Wannier functions 22. Polarization, localization and Berry's phases 23. Locality and linear scaling O (N) methods 24. Where to find more Appendixes References Index.

Electronic Structure: Basic Theory and Practical Methods

The quantum mechanics of proper open systems yields the physics that governs the local behavior of the electron density, ρ(r). The Ehrenfest force F(r) acting on an element of ρ(r) and the virial field ν(r) that determine its potential energy are obtained from equations of motion for the electronic momentum and virial operators, respectively. Each is represented by a “dressed” density, a distribution in real space that results from replacing the property in question for a single electron with a corresponding density that describes its average interaction with all of the remaining particles in the system. All bond paths, lines of maximum density linking neighboring nuclei in a system in stable electrostatic equilibrium, have a common physical origin in terms of F(r) and ν(r), regardless of the nature of the interaction. Each is homeomorphically mirrored by a virial path, a line of maximally negative potential energy density linking the same nuclei. The presence of a bond path and its associated virial path...

A Bond Path: A Universal Indicator of Bonded Interactions

The theory of molecular structure determined by the gradient vector field of the charge density ρ identifies the set of atomic interactions present in a molecule. The interactions so defined are characterized in terms of the properties of the Laplacian of the charge density ∇2ρ(r). A scalar field is concentrated in those regions of space where its Laplacian is negative and depleted in those where it is positive. An expression derived from the quantum mechanical stress tensor relates the sign of the Laplacian of ρ to the relative magnitudes of the local contributions of the potential and kinetic energy densities to their virial theorem averages. By obtaining a map of those regions where ∇2ρ(r) 0. The mechanics are characterized by the relatively large value of the kinetic energy, particularly the component parallel to the interaction line. In the closed‐shell interactions, the regions of dominant potential energy contributions are separately localized within the boundaries of each of the interacting atoms or molecules. In the shared interactions, a region of low potential energy is contiguous over the basins of both of the interacting atoms. The problem of further classifying a given interaction as belonging to a bound or unbound state of a system is also considered, first from the electrostatic point of view wherein the regions of charge concentration as determined by the Laplacian of ρ are related to the forces acting on the nuclei. This is followed by and linked to a discussion of the energetics of interactions in terms of the regions of dominant potential and kinetic energy contributions to the virial as again determined by the Laplacian of ρ. The properties of the Laplacian of the electronic charge thus yield a unified view of atomic interactions, one which incorporates the understandings afforded by both the Hellmann–Feynman and virial theorems.

The characterization of atomic interactions

Aus der Schrodingerschen Gleichung last sich durch eine kurze elementare Rechnung ohne Veruachlassigung die Beziehung

$$m\frac{{d^2 }}{{dt^2 }}\smallint \smallint \smallint d\tau .\Psi \Psi * .x = \smallint \smallint \smallint d\tau .\Psi \Psi * \left( { - \frac{{\partial V}}{{\partial x}}} \right)$$

ableiten, die fur ein kleines und klein bleibendes Wellenpaket (m von der Ordnung 1 g) besagt, das die Beschleunigung seiner Lagekoordinaten im Sinne der Newtonschen Bewegungsgleichungen zur ortlichen Kraft -∂V/∂x past.

Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik

The Conceptual Foundations of the Statistical Approach in Mechanics

In the case of a gravitating mass of perfect fluid which has come to thermodynamic equilibrium, it has previously been shown that the proper temperature T0 as measured by a local observer would depend in a definite manner on the gravitational potential at the point where the measurement is made. In the present article the conditions of thermal equilibrium are investigated in the case of a general static gravitational field which could correspond to a system containing solid as well as fluid parts. Writing the line element for the general static field in the form ds2=gijdxidxj+g44dt2 i,j=1,2,3,

where the gij and g44 are independent of the time t it is shown that the dependence of proper temperature on position at thermal equilibrium is such as to make the quantity T0sqrt[g44] a constant throughout the system.

/pdf/temperature-equilibrium-in-a-static-gravitational-field-1znma4ti9m.pdf

Temperature equilibrium in a static gravitational field

Der vorliegende Artikel steht in enger Beziehung zu V 8: L. Boltzmann und J. Nabl (Kinetische Theorie der Materie): Beide Artikel beschaftigen sich mit der Anwendung der Methoden der Wahrscheinlichkeitsrechnung auf das Studium der Bewegungen eines Molekulsystems. Wahrend sich aber V 8 vornehmlich den physikalischen Resultaten zuwendet, handelt es sich hier um die begrifflichen Grundlagen des Verfahrens.

P. Ehrenfest

Papers

Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik

The Conceptual Foundations of the Statistical Approach in Mechanics

Temperature equilibrium in a static gravitational field

Begriffliche Grundlagen der Statistischen Auffassung in der Mechanik

Adiabatische Invarianten und Quantentheorie