University of Minnesota-Duluth
The focus of my research is theoretical and computational hadronic physics. The computational work is aimed at the development of methods for the nonperturbative solution of quantum chromodynamics (QCD), in a form that, contrary to lattice gauge theory, will yield wave functions from which hadronic properties can be computed directly. Much of the theoretical work involves applications of perturbative QCD to scattering processes involving nuclei, particularly the deuteron.
The computational work is based on a Hamiltonian approach in the context of light-front coordinates, where t+z/c plays the role of time. This coordinate choice is driven by several important advantages, including boost invariance of the wave functions that describe any composite object, triviality of the vacuum state, and separation of internal and external momenta. It leads naturally to a Hamiltonian formulation in momentum space, and, given the simple vacuum, has a well-defined expansion in momentum eigenstates. Various theories have been considered: Yukawa theory (with S.J. Brodsky [SLAC} and G. McCartor [SMU]), quantum electrodynamics (QED; with S. Chabysheva [UMD], Brodsky, and McCartor), supersymmetric gauge theories (with S.S. Pinsky, U. Trittmann, N. Salwen, M. Harada, and Y. Proestos [Ohio State]), and phi4 theory (with Chabysheva). In this approach, the field-theoretic bound-state problem is reduced to a set of coupled integral equations for wave functions (of momentum) that describe the eigenstate. The eigenvalue is the square of the mass. The coupled system can be converted to a matrix eigenvalue problem by discretization. The most recent progress has been in the development of a new, light-front coupled-cluster method, that uses the mathematics of the standard coupled-cluster method to avoid truncations in particle number, and of a quantization scheme for QED that allows calculations in an arbitrary covariant gauge.
Hadronic scattering processes are conveniently treated by factorization into a hard scattering part, that can be treated as perturbative, and a soft part, that involves nonperturbative wave functions. The soft part is common to many different processes, which allows a systematic comparison of ratios in terms of only a perturbative analysis. With various collaborators, I have considered deuteron and 3He photodisintegration, pion photoproduction, hadronic form factors, and bounds on measures of proton structure. Chabysheva and I have recently completed a calculation of helicity amplitudes for deuteron photodisintegration, in order to understand the cross section for disintegration of a polarized target, and have been working on extensions to model form factors and electrodisintegration. We also plan to investigate the structure of nucleon form factors in the time-like region near threshold.
Another perturbative calculation is a collaboration with Chabysheva, Brodsky, and G.F. de Teramond [Costa Rica], on the use of the Lippmann-Schwinger (L-S) equation for an iterative improvement of the holographic QCD quark model. The idea is to use the difference between the QCD interaction and the model interaction as the driving term in the iteration of the L-S equation and thereby improve the model with inclusion of QCD effects. As a test of the idea, the analogous problem for muonium in QED is being investigated. A model equation for muonium, such as the Breit equation, is used as a starting point, and the L-S equation is used to introduce additional QED effects. It is anticipated that for both QED and QCD the calculation will eventually be storage limited, because the number of constituents included in the basis will rapidly expand with each iteration. Compression algorithms will be useful, and an adaptation of an algorithm by M. Weinstein et al., which uses singular value decomposition, has been derived for this purpose.
I was a member of the Consortium for Upper-Level Physics Software ( CUPS), which was brought together to develop computer programs useful in physics instruction. Approximately 30 physicists from English-speaking countries were involved. Funding was derived from the National Science Foundation, IBM, and Apple Computer. For each of nine physics courses frequently offered to juniors and seniors, there was formed a software development team to produce at least six programs, along with a supplementary text that provides documentation of the programs and discusses the associated physics. I belonged to two teams, Quantum Mechanics and Modern Physics, and produced four programs and four corresponding book chapters. The programs of the Quantum Mechanics team were voted best overall by the members of the consortium. The books and programs have been published by Wiley.
To download a patch for the Turbo Pascal runtime error 200 (caused by running on a PC with a clock faster than 200 MHz) go to http://www.pcmicro.com/elebbs/faq/rte200.html.
Quantum Mechanics Simulations (0-471-54884-7)
John Hiller, University of Minnesota-Duluth
Ian Johnston, University of Sydney (Australia)
Daniel Styer, Oberlin College
Contents: Bound State Wave Functions in One and Three Dimensions, Stationary Scattering States in One and Three Dimensions, Electron States on a Lattice, Quantum Mechanical Time Development, Identical Particles, and Bound States in Cylindrically Symmetric Potentials.
Modern Physics Simulations (0-4711-54882-0)
Douglas Brandt, Eastern Illinois University
John Hiller, University of Minnesota-Duluth
Michael Moloney, Rose-Hulman Institute
Contents: Historic Experiments in Electron Diffraction, Laser Cavities and Dynamics, Classical Scattering, Nuclear Properties and Decays, Special Relativity, Quantum Mechanics, and Hydrogen Atom and the H_2^+ Molecule.
Scattr1D solves the time-independent Schrodinger equation for stationary scattering states in one-dimensional potentials. The wave function is displayed in a variety of ways, and the transmission and reflection probabilities are computed. The probabilities may be displayed as functions of energy. The computations are done by numerically integrating the Schrodinger equation from the region of the transmitted wave, where the wave function is known up to some overall normalization and phase, to the region of the incident wave. There the reflected and incident waves are separated. The potential is assumed to be zero in the region of incidence and constant in the region of transmission.
Scattr3D performs a partial-wave analysis of scattering from a spherically symmetric potential. Radial and three-dimensional wave functions are displayed, as are phase shifts, and differential and total cross sections. The analysis employs an expansion in the natural angular momentum basis for the scattering wave function. The radial wave functions are computed numerically; outside the region where the potential is important they reduce to a linear combination of Bessel functions which asymptotically differs from the free radial wave function by only a phase. Knowledge of these phase shifts for the dominant values of angular momentum is used to approximate the cross sections.
CylSym solves the time-independent Schrodinger equation Hu=Eu in the case of a cylindrically symmetric potential for the lowest state of a chosen parity and magnetic quantum number. The method of solution is based on evolution in imaginary time, which converges to the state of the lowest energy that has the symmetry of the initial guess. The Alternating Direction Implicit method is used to solve a diffusion equation given by HU=-hbar dU/dt, where H is the Hamiltonian that appears in the Schrodinger equation. At large times, U is nearly proportional to the lowest eigenfunction of H, and the expectation value of H is an estimate for the associated eigenenergy.
Hatom computes eigenfunctions and eigenenergies for hydrogen, hydrogenic atoms, and single-electron diatomic ions. Hydrogenic atoms may be exposed to uniform electric and magnetic fields. Spin interactions are not included. The magnetic interaction used is the quadratic Zeeman term; in the absence of spin-orbit coupling, the linear term adds only a trivial energy shift. The unperturbed hydrogenic eigenfunctions are computed directly from the known solutions. When external fields are included, approximate results are obtained from basis-function expansions or from Lanczos diagonalization. In the diatomic case, an effective nuclear potential is recorded for use in calculation of the nuclear binding energy.
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