The conservation of quasiparticles, electric charge, and energy in weak-couplihg superconductors is discussed and correlated with the interpretation of the quasiparticles as particle and hole excitations. The formulation employed is that of the Bogolyubov-de Gennes equations. In the absence of spin-dependent interactions these equations are written in the form of a time-dependent Schrodinger equation with a two-component wave function and a two-by-two matrix Hamiltonian. Accordingly, the desired conservation laws are discussed on much the same basis as in ordinary quantum theory. The generalization to permit nonlocal quasiparticle and pair potentials and spin-dependent interactions is briefly presented.Die Konservierung von Quasipartikeln, elektrischer Ladung und Energie in schwach-gekoppelten Supraleitern wird besprochen und mit der Auslegung von Quasipartikeln als Partikel-und Lijcheranregungen in ffbereinstimmung gebracht. Die angewandten Formulierungen sind diejenigen der Bogolyubov-de Gennes-Gleichungen. Ohne spimbhiingige Wechselwirkungen werden diese Gleichungen als eine zeitabhiingige Schrodinger-Gleichung betrachtet, die als Wellenfunktion mit zwei Komponenten und einer Hamilton-Matrix vom Rang awei behandelt werden. Entsprechend werden die Gesetze der Konservierung auf der gleichen Basis wie in der iiblichen Quantentheorie besprochen. Die Verallgemeinerung auf nichtlokale Quasipartikelpotentiale und Paarpotentiale und auf spinabhiingige Wechselwirkungen wird kura besprochen.

We present a discussion of the energy associated with a one-dimensional mechanical wave which has a small amplitude but is otherwise general. We consider the kinetic energy only briefly because the standard treatments are adequate. However, our treatment of the potential energy is substantially more general and complete than the treatments which appear in introductory and intermediate undergraduate level physics textbooks. Specifically, we present three different derivations of the potential energy density associated with a one-dimensional, small amplitude mechanical wave. The first is based on the ‘‘virtual displacement’’ concept. The second is based on the ideas of stress and strain as they are generally used in dealing with the macroscopic elastic properties of matter. The third is based on the principle of conservation of energy, and also leads to an expression for the energy flux of the wave. We also present an intuitive and physical discussion based on the analogy between our system and a spring.

The concepts of energy density and energy current density for a single particle in nonrelativistic quantum theory are discussed. It is shown that the energy density ε and the energy current density Q are related by the equation (δε/δt)+∇·Q = S, where S represents external power sources.

The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and M ≡ z1-bM(1+a-b, 2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation.

We present a comprehensive discussion of the formulation of the kinematics of special relativity, i.e., the Lorentz transformation. We begin with a concise new proof that the principle of relativity implies that the transformation of event coordinates between inertial reference frames is linear. We then give a clear derivation of the pre-Lorentz transformation, which follows from the principle of relativity. We then show that the pre-Lorentz transformation and the inertial invariance of the speed of light together result in the Lorentz transformation. This, of course, is essentially the traditional formulation. We next present two additional formulations, one using Lorentz–Fitzgerald contraction and one using time dilation, instead of inertial invariance. This is reasonable since Lorentz–Fitzgerald contraction and time dilation are about as well established as and are arguably less abstract than inertial invariance, and thus may profitably be used instead of inertial invariance to complete the formulation. We then present a complete proof that the pre-Lorentz transformation and the requirement of closure upon composition together imply that the transformation is either a Galilean transformation or a generalized Lorentz transformation. This is noteworthy in that it gets ever so close to the Lorentz transformation without invoking light. In the course of this, we obtain a generalized velocity addition rule, which reduces to the velocity addition rule of special relativity. We next show that the generalized Lorentz transformation, together with inertial invariance, Lorentz–Fitzgerald contraction, and time dilation, used one at a time, yields three more formulations. We then show that the unspecified, nonzero, constant speed in the generalized Lorentz transformation can be determined without any reference to light, thereby obtaining a seventh formulation. Light plays no explicit role in the four formulations employing Lorentz–Fitzgerald contraction and time dilation and plays no role whatsoever in the seventh formulation. Thus, and this is a fact which should be strongly emphasized, the formulation of special relativity in no way depends upon the nature of electromagnetic radiation. We conclude by briefly discussing these seven formulations of the kinematics of special relativity and some associated implications.