Wesley E. Brittin

Liquid Rise in a Capillary Tube

A theory of the dynamics of capillary rise is developed by making certain assumptions as to the nature of the motion of the liquid in the tube. The most important assumptions are that the same forces act on the liquid when it is in an accelerated state of motion as when it is in a steady state, that the surface tension is constant, that the angle of contact between the meniscus of the liquid and the tube wall is constant, and that the wetting of the tube is not a rate‐determining factor of the motion. This theory leads to a second‐order non‐linear differential equation, the solution of which represents the motion of the liquid in the tube. A formal solution of the differential equation is obtained in the form of a double Dirichlet series. Approximations to the series are compared with experimental data, and it is concluded that the agreement between theory and experiment is satisfactory.

Poincaré gauge in electrodynamics

The gauge presented here, which we call the Poincare gauge, is a generalization of the well‐known expressions φ = −r⋅E0 and A = 1/2 B0×r for the scalar and vector potentials which describe static, uniform electric and magnetic fields. This gauge provides a direct method for calculating a vector potential for any given static or dynamic magnetic field. After we establish the validity and generality of this gauge, we use it to produce a simple and unambiguous method of computing the flux linking an arbitrary knotted and twisted closed circuit. The magnetic flux linking the curve bounding a Mobius band is computed as a simple example. Arguments are then presented that physics students should have the opportunity of learning early in their curriculum modern geometric approaches to physics. (The language of exterior calculus may be as important to future physics as vector calculus was to the past.) Finally, an appendix illustrates how the Poincare gauge (and others) may be derived from Poincare’s lemma relatin...

Number operators for composite particles in nonrelativistic many-body theory

Commuting physical occupation number operators for composite particles are constructed using projection operator techniques. The composite particle occupation number operators are constructed from creation and annihilation operators of the elementary particles which make up the many‐body system. They appear as positive operators in any given second quantized theory and represent observables within the framework of that theory. Bose‐type composites have number operators with eigenvalues 0,1,2,..., and Fermi‐type composites have number operators with eigenvalues 0,1. There does not arise here any problem having to do with exchange symmetry—exchange symmetry is exact, since the number operators act in the Fock space of the elementary particles. The composite particle number operators may be used in the construction of theories of composite particle reactions or equilibrium from a first principles standpoint. The construction used here not only establishes the existence of composite particle number operators ...

Taylor's theorem for analytic functions of operators

We discuss analytic functions on a Banach algebra into itself. In particular expressions for derivatives are given as well as convergent Taylor expansions.

Functional Methods in Statistical Mechanics. I. Classical Theory

A statistical‐mechanical theory of fields is developed. Since a field has an infinite number of degrees of freedom, it is natural and convenient to use functional methods for its description. The most general statistical‐mechanical state for a field is represented by a distribution functional which satisfies a functional differential equation analogous to the Liouville equation. The functional Fourier transform (characteristic functional) is introduced and its properties are studied. Multitime functionals and various reduced distribution functions are also discussed. The formalism is applied to the free electromagnetic fields as well as to a system of charged particles (plasma) interacting via the electromagnetic field.

Projection operator techniques in physics

A systematic account of projection operators (projectors) and orthogonalization techniques together with applications to selected areas of physics is presented. This unified approach is shown to have advantages over other approaches in that the mathematical statements are more precise. The mathematical level, however, is aimed at the practicing physicist and lies between rigorous mathematics and current use in physics. Further, the techniques presented have practical applications as is demonstrated by examples in the quantum theory of measurement, in the relationship between second quantization and configuration space techniques, and in an account of generalized Wannier and Bloch functions. Attention is paid to the problem of construction of orthogonal projection operators (orthogonal projectors). The construction of orthogonal projectors even in approximate form would allow the solution of many practical problems ranging from the eigenvalue spectrum problem to the construction of states for many‐body sys...

The interaction of radiation with charged particles. — I

SummaryWe employ the Bogoliubov-Tyablikov transformation and an equation of motion method, utilizing part of the Hamiltonian for a system of interacting electrons and photons, to obtain the familiar dispersion relation, ω2=k2c2+ωp2, for a system of quasi-photons. The equation of motion method can be used to obtain an extended dispersion relation when more terms of the Hamiltonian are taken into account.RiassuntoServendoci di parte dell’hamiltoniano per un sistema di elettroai e fotoni inter-agenti, per ottenere la nota relazione di dispersione ω2 =k2c2 + ω per un sistema di quasi-fotoni utilizziamo la trasformazione di Bogoliubov-Tyablinov e un’equazione di moto. Il metodo dell’equazione di moto si può usare per ottenere una relazione di dispersione estesa quando si tenga conto di più termini dell’hamiltoniano.

Mixed boundary value problems for concentric cylinders—an algebraic operator formulation

Unitary operators are introduced which act on the Fourier transforms of the boundary values on concentric cylinders of a harmonic function in E 3. The boundary values on the concentric cylinders are determined from given mixed boundary conditions on the same cylinders, and the solution for them is expressed explicitly and simply through the use of these unitary operators plus certain other self adjoint positive operators. The boundary values on the cylinders then determine the harmonic function everywhere in E 3.

Composite Particles in Many-Body Theory

The stability of free atoms and molecules (and even interacting atoms and molecules, if the interactions are not too large) allows us to consider them as “particles” with structure. The successes of early kinetic theory and spectroscopy bear this out. We know in fact that atoms, molecules, nuclei, etc. may be thought of as being composed of particles more elementary than themselves. Of course the nuclei which are “elementary” for atoms and molecules are “composite” in terms of protons and neutrons. Composite structures when far enough apart or not interacting too strongly obey Fermi (Bose) statistics if they are themselves made of odd (even) numbers of fermions. This is just a statement of the famous Ehrenfest-Oppenheimer theorem. We present a method for treating interacting composites as though they were particles obeying simple Bose or Fermi commutation relations. Our method appears to generalize and unify a number of previous methods.

Uncertainty Principle and Normal Operators—Some Comments

It is shown that an appropriate formulation of the uncertainty principle for normal operators leads to the same mathematical expression as that for Hermitian operators. The treatment is valid for mixed states as well as for pure states.