vii 3.3.1 Difficulties related to the classical bound state and the quantum bound and scattering states..

No abstract

In this work, we prove a previously published conjecture that a prescription we gave for constructing states that implement Gauss's law for 'pure glue' QCD is correct. We also construct a unitary transformation that extends this prescription so that it produces additional states that implement Gauss's law for QCD with quarks as well as gluons. Furthermore, we use the mathematical apparatus developed in the course of this work to construct gauge-invariant spinor (quark) and gauge (gluon) field operators. We adapt this SU (3) construction for the SU (2) Yang-Mills case, and we consider the dynamical implications of these developments.

We calculate the Wigner quasi-probability distribution for position and momentum, P (n) W (x, p), for the energy eigenstates of the standard infinite well potential, using both x-and p-space stationary-state solutions, as well as visualizing the results. We then evaluate the time-dependent Wigner distribution, P W (x, p; t), for Gaussian wave packet solutions of this system, illustrating both the short-term semi-classical time dependence, as well as longer-term revival and fractional revival behavior and the structure during the collapsed state. This tool provides an excellent way of demonstrating the patterns of highly correlated Schrödinger-cat-like 'mini-packets' which appear at fractional multiples of the exact revival time.

We use a unitary operator constructed in earlier work to transform the Hamiltonian for QCD in the temporal ($A_0=0$) gauge into a representation in which the quark field is gauge-invariant, and its elementary excitations -- quark and antiquark creation and annihilation operators -- implement Gauss's law. In that representation, the interactions between gauge-dependent parts of the gauge field and the spinor (quark) field have been transformed away and replaced by long-range non-local interactions of quark color charge densities. These long-range interactions connect SU(3) color charge densities through an infinite chain of gauge-invariant gauge fields either to other SU(3) color charge densities, or to a gluon "anchor". We discuss possible implications of this formalism for low-energy processes, including confinement of quarks that are not in color singlet configurations.Comment: 9 pages, LaTeX, uses REVTe

We construct a transformation that transforms perturbative states into states that implement Gauss's law for `pure gluonic' Yang-Mills theory and QCD. The fact that this transformation is not and cannot be unitary has special significance. Previous work has shown that only states that are unitarily equivalent to perturbative states necessarily give the same S-matrix elements as are obtained with Feynman rules.Comment: 11 page

We derive new constraints on the zeros of Airy functions by using the so-called quantum bouncer system to evaluate quantum-mechanical sum rules and perform perturbation theory calculations for the Stark effect. Using commutation and completeness relations, we show how to systematically evaluate sums of the form S p (n) = k =n 1/(ζ k − ζ n ) p , for natural p > 1, where −ζ n is the n th zero of Ai(ζ).

We discuss special k = 2m(E − V (x))/ 2 = 0 (i. e. zero-curvature) solutions of the onedimensional Schrödinger equation in several model systems which have been used as idealized versions of various quantum well structures. We consider infinite well plus Dirac delta function cases (where E = V (x) = 0) and piecewise-constant potentials, such as asymmetric infinite wells (where E = V (x) = V 0 > 0). We also construct supersymmetric partner potentials for several of the zero-energy solutions in these cases. One application of zero-curvature solutions in the infinite well plus δ-function case is the construction of 'designer' wavefunctions, namely zero-energy wavefunctions of essentially arbitrary shape, obtained through the proper placement and choice of strength of the δ-functions.