Kramer, T., Heller, E.J. & Parrott, R.E. An efficient and accurate method to obtain the energy-dependent Green function for general potentials.

Journal of Physics: Conference Series 99, 012010 (2008).

Publisher's VersionAbstractTime-dependent quantum mechanics provides an intuitive picture of particle propagation in external fields. Semiclassical methods link the classical trajectories of particles with their quantum mechanical propagation. Many analytical results and a variety of numerical methods have been developed to solve the time-dependent Schrödinger equation. The time-dependent methods work for nearly arbitrarily shaped potentials, including sources and sinks via complex-valued potentials. Many quantities are measured at fixed energy, which is seemingly not well suited for a time-dependent formulation. Very few methods exist to obtain the energy-dependent Green function for complicated potentials without resorting to ensemble averages or using certain lead-in arrangements. Here, we demonstrate in detail a time-dependent approach, which can accurately and effectively construct the energy-dependent Green function for very general potentials. The applications of the method are numerous, including chemical, mesoscopic, and atomic physics.

Wasserman, A., Maitra, N.T. & Heller, E.J. Investigating interaction-induced chaos using time-dependent density-functional theory.

Physical Review A 77, 042503 (2008).

Publisher's VersionAbstractSystems whose underlying classical dynamics are chaotic exhibit signatures of the chaos in their quantum mechanics. We investigate the possibility of using the linear response formalism of time-dependent density functional theory (TDDFT) to study the case when chaos is induced by electron-interaction alone. Nearest-neighbor level-spacing statistics are in principle exactly and directly accessible from TDDFT. We discuss how the TDDFT linear response procedure can reveal information about the mechanism of chaos induced by electron-interaction alone. A simple model of a two-electron quantum dot highlights the necessity to go beyond the adiabatic approximation in TDDFT.

Heller, E.J. & Tomsovic, S. Postmodern quantum mechanics.

Physics Today 46, 38–46 (2008).

Publisher's VersionAbstract

Postmodern movements are well known in the arts. After a major artistic revolution, and after the “modern” innovations have been assimilated, the threads of premodern thought are always reconsidered. Much of value may be rediscovered and put to new use. The modern context casts new light on premodern thought, which in turn shades perspectives on modernism.

Recent progress in semiclassical theory has overcome barriers posed by classical chaos and cast light on the correspondence principle. Semiclassical ideas have also become central to new experiments in atomic, molecular, microwave and mesoscopic physics.

Heller, E.J. Surface physics: Electron wrangling in quantum corrals. Nature Physics 4, 443 - 444 (2008).

Publisher's VersionAbstractThe article reports on the discovery of image electron waves' ability to propagate along a metal surface by Don Eigler, Christopher Lutz, and Michael Crommie at IBM Almaden in 1990. Eigler, Lutz, and Crommie noticed unexpectedly long and periodic undulations in the scanning tunnelling microscope (STM) conductance signal as a function of tip position. The undulations were recognized as the signature of surface confined electron waves whose de Broglie wavelength is bigger than the lattice.

Heller, E.J., Kaplan, L. & Pollmann, F. Inflationary dynamics for matrix eigenvalue problems. Proceedings of the National Academy of Sciences of the United States of America 105, 7631 - 7635 (2008).

Publisher's VersionAbstractMany fields of science and engineering require finding eigenvalues and eigenvectors of large matrices. The solutions can represent oscillatory modes of a bridge, a violin, the disposition of electrons around an atom or molecule, the acoustic modes of a concert hall, or hundreds of other physical quantities. Often only the few eigenpairs with the lowest or highest frequency (extremal solutions) are needed. Methods that have been developed over the past 60 years to solve such problems include the Lanczos algorithm. Jacobi-Davidson techniques, and the conjugate gradient method. Here, we present a way to solve the extremal eigenvalue/eigenvector problem, turning it into a nonlinear classical mechanical system with a modified Lagrangian constraint. The constraint induces exponential inflationary growth of the desired extremal solutions. [ABSTRACT FROM AUTHOR]