R. W. Robinett

Quantum wave packet revivals

Abstract The numerical prediction, theoretical analysis, and experimental verification of the phenomenon of wave packet revivals in quantum systems has flourished over the last decade and a half. Quantum revivals are characterized by initially localized quantum states which have a short-term, quasi-classical time evolution, which then can spread significantly over several orbits, only to reform later in the form of a quantum revival in which the spreading reverses itself, the wave packet relocalizes, and the semi-classical periodicity is once again evident. Relocalization of the initial wave packet into a number of smaller copies of the initial packet (‘minipackets’ or ‘clones’) is also possible, giving rise to fractional revivals. Systems exhibiting such behavior are a fundamental realization of time-dependent interference phenomena for bound states with quantized energies in quantum mechanics and are therefore of wide interest in the physics and chemistry communities. We review the theoretical machinery of quantum wave packet construction leading to the existence of revivals and fractional revivals, in systems with one (or more) quantum number(s), as well as discussing how information on the classical period and revival time is encoded in the energy eigenvalue spectrum. We discuss a number of one-dimensional model systems which exhibit revival behavior, including the infinite well, the quantum bouncer, and others, as well as several two-dimensional integrable quantum billiard systems. Finally, we briefly review the experimental evidence for wave packet revivals in atomic, molecular, and other systems, and related revival phenomena in condensed matter and optical systems.

QUANTUM ELECTRODYNAMICS AT FINITE TEMPERATURE

Abstract We present a systematic examination of finite temperature effects in quantum electrodynamics at one loop order. We calculate mass, charge, and wavefunction renormalization, demonstrate the running of the coupling constant at high temperatures, and study the renormalized vertex function and the energy momentum tensor. The confusion in the literature concerning the finite temperature corrections to the electrons magnetic moment is resolved. We also present the finite temperature effects in scalar electrodynamics. Throughout we stress the need to provide a well-defined method to observe a given quantity when interpreting the results of a calculation, and we suggest new techniques which incorporate the novel features of finite temperature theories.

Visualizing the collapse and revival of wave packets in the infinite square well using expectation values

We investigate the short-, medium-, and long-term time dependence of wave packets in the infinite square well. In addition to emphasizing the appearance of wave packet revivals, i.e., situations where a spreading wave packet reforms with close to its initial shape and width, we also examine in detail the approach to the collapsed phase where the position-space probability density is almost uniformly spread over the well. We focus on visualizing these phenomena in both position- and momentum-space as well as by following the time-dependent expectation values of and uncertainties in position and momentum. We discuss the time scales for wave packet collapse, using both an autocorrelation function analysis as well as focusing on expectation values, and find two relevant time scales which describe different aspects of the decay phase. In an Appendix, we briefly discuss wave packet revival and collapse in a more general, one-dimensional power-law potential given by V(k)(x)=V0|x/a|k which interpolates between th...

Production of gauge-fermions at colliders☆

Abstract The gauge fermions of supersymmetry can be produced at p p, pp and e + e − colliders, via the exchange of supersymmetric scalar particles and via W ± , γ and Z 0 direct channel diagrams. We give expressions for amplitudes of ggω + and ω + ω − production in quark-antiquark collisions and ω + ω − production in e + e − collisions, where gg and ω + are gauge fermion mass eigenstates. In the supersymmetry limit gauge amplitude zeros appear in the u d →ggω + amplitudes. Integrated rates are given for a light gg and representative w and scalar masses.

Visualizing classical periodic orbits from the quantum energy spectrum via the Fourier transform: Simple infinite well examples

The Fourier transform of the density of quantized energy levels for a quantum mechanical particle in a two-dimensional (2-D) infinite well (or billiard geometry) is known to exhibit δ-function-like spikes at distance values (L) corresponding to the lengths of periodic orbits or closed trajectories. We show how these Fourier transforms can be rather easily calculated numerically for simple infinite well geometries including the square and rectangular well in 2 D, the cubical well in three dimensions, as well as the circular infinite well (and variations) in two dimensions. Such calculations provide a novel, well-motivated, and relatively straightforward example of numerical Fourier transform techniques and make interesting connections between quantum energy levels and classical trajectories in a way which is seldom stressed in the undergraduate curriculum.

Wigner quasi-probability distribution for the infinite square well: Energy eigenstates and time-dependent wave packets

We calculate and visualize the Wigner quasi-probability distribution for the position and momentum, PW(n)(x,p), for the energy eigenstates of the infinite square well. We evaluate the time-dependent Wigner distribution, PW(x,p;t), for Gaussian wave packet solutions of this system, and illustrate the short-term semi-classical time dependence and the longer-term revival and fractional revival behavior. Our results indicate how the Wigner distribution can be used to examine the highly correlated dynamical position-momentum structure of quantum states. In particular, this tool provides an excellent way of demonstrating the patterns of highly correlated Schrodinger-cat-like “mini-packets” which appear at fractional multiples of the exact revival time.

Periodic orbit theory analysis of the circular disk or annular billiard: Nonclassical effects and the distribution of energy eigenvalues

Periodic orbit (PO) theory can be used to make connections between the quantum energy eigenvalue spectrum and the closed orbits of the corresponding classical system. The two-dimensional annular billiard or circular disk system (namely, a particle in the plane confined between inner and outer infinite circular walls at fR≡Rin<Rout≡R where 0<f<1) is examined in the context of periodic orbit theory to illustrate several novel aspects of the PO analysis. Important features of this problem include (i) the appearance and disappearance of various features in the classical path length spectrum as a parameter (in this case f=Rin/Rout), is continuously varied, (ii) the presence of path length features which do not correspond to classical trajectories, but are rather due to purely wavelike phenomena (namely, diffraction around the inner annulus), and (iii) the study of the contribution of different regions in the energy eigenvalue space to different classes of classical trajectories. This last feature is a general ...

Low transverse momentum ψ and production in polarized proton-proton collisions

Abstract We present calculations of the spin-spin asymmetry in the low transverse momentum production of ψs and s in polarized proton-proton collisions. A perturbative QCD approach using quarkonium wavefunctions derived from non-relativistic potential models is employed. The contributions from gg→ χ 0,2 and gg→g 3 S 1 are known to dominate in this approach and we exhibit the sensitivity of the polarization asymmetry to various choices of polarized gluon distribution.

Wave packet construction in two-dimensional quantum billiards: Blueprints for the square, equilateral triangle, and circular cases

We present quasianalytical and numerical calculations of Gaussian wave packet solutions of the Schrodinger equation for two-dimensional infinite well and quantum billiard problems with equilateral triangle, square, and circular footprints. These cases correspond to N=3, N=4, and N→∞ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wave functions are given in terms of familiar special functions. For the first two systems, we obtain closed form expressions for the expansion coefficients for localized Gaussian wave packets in terms of the eigenstates of the particular geometry. For the circular case, we discuss numerical approaches. We use these results to discuss the short-time, quasiclassical evolution in these geometries and the structure of wave packet revivals. We also show how related half-well problems can be easily solved in each of the three cases.

Limits on the ? neutrino electromagnetic properties from single photon searches ate + e ? colliders

AbstractWe set limits on the magnetic moment and charge radius of the τ neutrino by examining the contributions to the processe+e−→v