John R. Hiller Professor, Department of Physics University of Minnesota-Duluth

• Contact Information
• Research Interests
• Research Publications (as listed by SPIRES)
• Course Web Server
• Instructional Software

Contact Information

Office: 353 Marshall W. Alworth Hall (MWAH)
Hours: 11 MWF and by appointment. E-mail: jhiller at d.umn.edu
Voice: 218-726-7594
Fax: 218-726-6942
WWW: http://www.d.umn.edu/~jhiller (This web page!)
Address:

Research Interests

The primary goal of my research is to develop computational methods for the nonperturbative solution of quantum chromodynamics (QCD). The approach that I have adopted is a Hamiltonian formalism based on light-cone quantization. Within this approach one has a well-defined Fock-state expansion and boost-invariant wave functions that, if known, would provide the basis for computation of hadronic properties. The wave functions satisfy an infinite system of coupled integral equations derived from the field-theoretic eigenvalue problem of the mass-squared operator. Once this system is solved, the wave functions can be used to compute form factors, cross sections for exclusive processes, and other quantities of interest, principally as matrix elements in the second-quantized hadronic state. This approach, with its direct focus on wave functions, is complementary to lattice QCD.

The system of equations is converted to a matrix eigenvalue problem by some form of discretization. The most commonly used tool is discrete light-cone quantization (DLCQ), which begins at the second-quantized level but is roughly equivalent to a trapezoidal approximation to integral operators in momentum space. The matrix representation of the mass-squared operator is very sparse, which has made it ideal for diagonalization by iterative methods such as the Lanczos algorithm. The largest basis sets used in my work have reached 60 million; this is more than an order of magnitude larger than any other field-theoretic calculation of this type.

Recent efforts, in collaboration with S. J. Brodsky of the Stanford Linear Accelerator Center and G. McCartor of Southern Methodist University, have been aimed at investigating the usefulness of Pauli-Villars regularization. Pauli-Villars particles are introduced directly into the Fock basis and coupled in ways that, at least in perturbation theory, will induce the cancellations necessary to regulate the theory. This approach has been successfully tested for two simple models, and is now being applied to a single-fermion truncation of Yukawa theory and QED.

I also do work on supersymmetric theories, as part of the SDLCQ Collaboration led by S.S. Pinsky of the Ohio State University. In this work we exploit a special supersymmetric form of DLCQ.

Instructional Software

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.brain.uni-freiburg.de/~klaus/pascal/runerr200.

Quantum Mechanics Simulations (0-471-54884-7)
Authors:
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)
Authors:
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.

Abstracts of Quantum Mechanics programs written by me:

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.

Abstract of the Modern Physics program written by me:

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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