scholarly journals The general proof of certain fundamental equations in the theory of metallic conduction

1934 ◽  
Vol 144 (851) ◽  
pp. 101-117 ◽  
Keyword(s):  
Wave Equation ◽  
Periodic Potential ◽  
Thermal Motion ◽  
Electronic Conduction ◽  
Modern Theory ◽  
General Proof ◽  
Metallic Conduction ◽  
Fundamental Equations

In the modern theory of electronic conduction the electrons are considered, when the thermal motion of the lattice is neglected, as moving in a periodic potential with the property V ( x + la , y + ma , z + na ) = V ( x, y, z ). The wave equation for an electron in this field is { h 2/8π2 m ∇ 2 + E K - V} ψ K = 0. Block has shown that this equation has solutions of the form ψ K = e i K.R U K (R), where U K has the periodicity of the lattice.

scholarly journals On the foundations of the electron wave equation

1933 ◽  
Vol 139 (838) ◽  
pp. 349-368 ◽  
Keyword(s):  
Wave Equation ◽  
Classical Mechanics ◽  
Uniform Motion ◽  
Modern Theory ◽  
Dimensional Manifold ◽  
Wave Mechanics ◽  
Straight Line ◽  
The World ◽  
The Subject ◽  
Historical Necessity

1. There are two problems which are fundamental to wave mechanics— that of deducing from first principles the properties of the wave field of ψ, by means of which, on modern theory, the mechanical properties of matter are to be described, and that of formulating a logical description of the properties of matter in terms of ψ. The former only of these problems is discussed in this communication. In spite of many valuable papers that have been published on the subject of Dirac’s remarkable wave equation, the derivation of the equation still seemed (to the present writer, at any rate,) to include some obscurities which could perhaps be removed. To some extent this is owing to the development of the equation, as a matter of historical necessity, having come about by successive extensions to classical mechanics; so that some difficulties have been produced by the fundamental relativity considerations having been introduced into the theory at a late stage, instead of in their proper place at the beginning. In what follows, by treating the problem from the beginning as a four-dimensional one, a deduction of the wave equation free from empirical steps is, I think, obtained, while also certain new features of the equation are brought to light. 2. The Principle of Action .—The method of four-dimensional mechanics is to assume that the motions of bodies in the world can be represented by “tracks,” or curved lines, in a “fourfold.” (This last term will be used here to mean a four-dimensional manifold with Euclidian geometry, i. e ., in which ds 2 = dx i 2 ( i = 1, . . . 4).) (1) Uniform motion in the world is represented by straight line tracks in the fourfold, and the “classical relativity” laws of motion are derived by formulating the simplest mathematical specification of curved tracks which will represent the non-uniform motions of bodies that are usually observed in the world.

Dispersed stable states spectrum of the wave equation with space-time periodic potential

2013 ◽  
Vol 103 (5) ◽  
pp. 50001
Author(s):  
B. S. Alexandrov ◽  
A. R. Bishop ◽  
N. Zahariev ◽  
I. Kostadinov
Keyword(s):  
Wave Equation ◽  
Periodic Potential ◽  
Space Time ◽  
Stable States ◽  
Time Periodic

scholarly journals The theory of the change in resistance in a magnetic field

1934 ◽  
Vol 145 (854) ◽  
pp. 268-277 ◽  
Keyword(s):  
Magnetic Field ◽  
Temperature Dependence ◽  
Periodic Potential ◽  
Quadratic Function ◽  
Modern Theory ◽  
Quantum Numbers ◽  
Wrong Direction ◽  
Constant Potential ◽  
The Subject ◽  
Experimental Values

The resistance of a metal is in general increased by a magnetic field. For sufficiently small magnetic fields this dependence may, of course, be expressed by the equation ∆R/R = BH 2 . The calculation of the coefficient B has been the subject of many previous investigations. Sommerfeld has shown that if the electrons are regarded as moving in a constant potential, then the theoretical value of B is 10,000 times smaller than the observed value, and the temperature dependence is in the wrong direction. Peierls has suggested that the correct magnitude of B may be obtained if cognizance is taken of the fact that the electrons are moving in a periodic potential, as is usual in the modern theory of metals. Moreover, he has shown that the correct temperature dependence will then be obtained. Blochinzev and Nordheim have recently investigated in detail the charge of resistance of divalent metals from this standpoint. The periodicity of the lattice was, however, introduced in quite an idealized manner. In place of an actual metal they considered a simple cubic lattice. The surface of the Fermi distribution was assumed to be composed of sections each of which was a quadratic function of the quantum numbers ξ, η , ζ. The coefficient B was obtained in terms of the Fourier coefficient of the potential energy of an electron in the lattice, V 100 . Agreement with the experimental values of B was obtained by taking |V 100 | as small as several hundredths of an electron volt, while the correct value must be of the order of 1 volt. Since in their model B varies inversely as the square of |V 100 |, the comparison with experiment is far from satisfactory.

scholarly journals A general kinetic theory of liquids III. Dynamical properties

1947 ◽  
Vol 190 (1023) ◽  
pp. 455-474 ◽  
Keyword(s):  
Kinetic Theory ◽  
Intermolecular Forces ◽  
Distribution Functions ◽  
Thermal Motion ◽  
Theory Of Liquids ◽  
Full Discussion ◽  
Molecular Forces ◽  
Monatomic Fluids ◽  
Fundamental Equations

The theory of liquids formulated in part I and applied to the equilibrium state in part II is here extended to liquids in motion. The connexion between the macroscopic and microscopic properties is revealed by the derivation of a set of generalized hydrodynamical equations, of which the fundamental equations of hydrodynamics are a special case; the more general equations describe the mean motion of clusters of molecules in the fluid. It is shown that the pressure tensor and energy-flux vector in a fluid consist of two parts, due to the thermal motion of the molecules and the intermolecular forces respectively, of which only the first is found in the kinetic theory of gases, but of which the second is dominant for the liquid state. A method is evolved for the study of those ‘normal' non-uniform states which relate to actual monatomic fluids in motion. It becomes apparent, as in the case of equilibrium, that there is a region of temperature and density where analytical singularities arise, closely associated with the process of condensation. Rigorous expressions for the coefficients of viscosity and thermal conduction are then derived which apply equally to the liquid and the gas. They consist of two parts due to the thermal motion and molecular forces respectively, of which the first is dominant for the gas, and the second for the liquid. By approximating to the rigorous formula, an expression for the viscosity of liquids is obtained, comparable with certain other formulae, previously proposed on quasi-empirical grounds, and giving good agreement with experiment. An integro-differential equation is derived for the determination of the distribution functions relating to the non-uniform state. A full discussion is given of the simplest case, and the velocity distribution in, non-uniform liquids and gases examined.

The classical field theory of matter and electricity - The electromagnetic theory of particles

1960 ◽  
Vol 253 (1025) ◽  
pp. 205-226 ◽  
Keyword(s):  
Field Theory ◽  
Wave Equation ◽  
Classical Field Theory ◽  
Relativity Theory ◽  
Modern Theory ◽  
Field Equations ◽  
Electromagnetic Field Theory ◽  
Energy And Momentum ◽  
Static Form ◽  
Classical Particles

The electromagnetic field theory developed in the previous paper is here applied to the problem of devising systems which behave as classical particles. It is found that spherically symmetrical systems can exist which, when they are stationary: (1) satisfy the static form of the extended equations at every point of space; and (2) are characterized mechanically by being everywhere in equilibrium under the sole action of the Maxwellian stress of their own field—thus they are pure electromagnetic systems subsisting free of external constraint. (3) When they are transformed so as to be in motion, the energy and momentum they possess are exactly those required for material particles by relativity theory. A rather obvious restriction made on the generality of the conditions for particle existence brought to light the possibility of a solution denoting an ‘atomic’ system built up of successive shells, each of which must contain the same energy, and net charge, as the others. The reason for such a result is that, when their very great generality is restricted in the most straightforward way, the field equations reduce to the form of a wave equation. The relation of this to the wave equation of modern theory is briefly discussed. The transformation behaviour of the field equations when a Lorentz transformation is applied to the co-ordinates is dealt with in this paper; it is found that they remain invariant in form under wider transformations of the field variables than are permitted by the classical equations. The variables may be submitted to a certain transformation without the co-ordinates being transformed at all. The physical meaning of this is investigated and an explanation of it found.

AN X-RAY DIFFRACTION STUDY OF THE THERMAL MOTION OF ATOMS IN β-TIN

1980 ◽  
Vol 41 (C1) ◽  
pp. C1-145-C1-146 ◽  
Author(s):  
B. Greenberg ◽  
G. M. Rothberg
Keyword(s):  
Diffraction Study ◽  
Thermal Motion ◽  
X Ray Diffraction ◽  
X Ray
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