arXiv subjects

R. W. Robinett

Publications and source records attributed to R. W. Robinett.

At least 37 records · Page 2Linked to original sources

arXiv2003

Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum

The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low- and high-energy limits respectively. The energy variation of the classical periodicity ($τ$) is also dramatic, having the special limiting case of $τ\to \infty$ at the 'top' of the classical motion (i.e. the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to 4th order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the 'top' in energy, from either above or below, the revival times decrease from their low- and high-energy values until very near the separatrix where they increase to a large, but finite value.

quant-ph↗

arXiv2003

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

We present quasi-analytical and numerical calculations of Gaussian wave packet solutions of the Schrödinger 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 $N \to \infty$ regular polygonal billiards and infinite wells, respectively. In each case the energy eigenvalues and wavefunctions 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 wavepackets 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, quasi-classical 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.

quant-ph↗

arXiv2003

Visualizing classical and quantum probability densities for momentum using variations on familiar one-dimensional potentials

After briefly reviewing the definitions of classical probability densities for position, $P_{CL}(x)$, and for momentum, $P_{CL}(p)$, we present several examples of classical mechanical potential systems, mostly variations on such familiar cases as the infinite well and the uniformly accelerated particle for which the classical distributions can be easily derived and visualized. We focus especially on a simple potential which interpolates between the symmetric linear potential, $V(x) = F|x|$, and the infinite well, which can illustrate, in a mathematically straightforward way, how the divergent, $δ$-function classical probability density for momentum for the infinite well can be easily seen to arise. Such examples can help students understand the quantum mechanical momentum-space wavefunction (and its corresponding probability density) in much the same way that other semi-classical techniques, such as the WKB approximation, can be used to visualize position-space wavefunctions.

quant-ph↗

arXiv2003

Quantum mechanical analysis of the equilateral triangle billiard: periodic orbit theory and wave packet revivals

Using the fact that the energy eigenstates of the equilateral triangle infinite well (or billiard) are available in closed form, we examine the connections between the energy eigenvalue spectrum and the classical closed paths in this geometry, using both periodic orbit theory and the short-term semi-classical behavior of wave packets. We also discuss wave packet revivals and show that there are exact revivals, for all wave packets, at times given by $T_{rev} = 9 μa^2/4\hbar π$ where $a$ and $μ$ are the length of one side and the mass of the point particle respectively. We find additional cases of exact revivals with shorter revival times for zero-momentum wave packets initially located at special symmetry points inside the billiard. Finally, we discuss simple variations on the equilateral ($60^{\circ}-60^{\circ}-60^{\circ}$) triangle, such as the half equilateral ($30^{\circ}-60^{\circ}-90^{\circ}$) triangle and other `foldings', which have related energy spectra and revival structures.

quant-ph↗

arXiv2003

Expectation value analysis of wave packet solutions for the quantum bouncer: short-term classical and long-term revival behavior

We discuss the time development of Gaussian wave packet solutions of the quantum bouncer' (a quantum mechanical particle subject to a uniform downward force, above an impermeable flat surface). We focus on the evaluation and visualization of the expectation values and uncertainties of position and momentum variables during a single quasi-classical period as well as during the long term collapsed phase and several revivals. This approach complements existing analytic and numerical analyses of this system, as well as being useful for comparison with similar results for the harmonic oscillator and infinite well cases.

quant-ph↗

arXiv2003

Visualizing the collapse and revival of wavepackets 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 $V_{(k)}(x) = V_0|x/a|^k$ which interpolates between the case of the harmonic oscillator ($k=2$) and the infinite well ($k=\infty$).

quant-ph↗

arXiv2003

Quantum mechanics of the two-dimensional circular billiard plus baffle system and half-integral angular momentum

We examine the quantum mechanical eigensolutions of the two-dimensional infinite well or quantum billiard system consisting of a circular boundary with an infinite barrier or baffle along a radius. Because of the change in boundary conditions, this system includes quantized angular momentum values corresponding to half-integral multiples of $\hbar/2$. We discuss the resulting energy eigenvalue spectrum and visualize some of the novel energy eigenstates found in this system. We also discuss the density of energy eigenvalues, $N(E)$, comparing this system to the standard circular well. These two billiard geometries have the same area (A=$πR^2$), but different perimeters ($P=2πR$ versus $(2π+ 2) R$), and we compare both cases to fits of $N(E)$ which make use of purely geometric arguments involving only $A$ and $P$. We also point out connections between the angular solutions of this system and the familiar pedagogical example of the one-dimensional infinite well plus $δ$-function potential.

quant-ph↗

arXiv2003

Quantum wave packet revivals in circular billiards

We examine the long-term time-dependence of Gaussian wave packets in a circular infinite well (billiard) system and find that there are approximate revivals. For the special case of purely $m=0$ states (central wave packets with no momentum) the revival time is $T_{rev}^{(m=0)} = 8μR^2/\hbar π$, where $μ$ is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that $T_{rev}^{(m \neq 0)} = (π^2/2) T_{rev}^{(m=0)} \approx 5T_{rev}^{(m=0)}$ and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of $m \neq 0$ states is/are increased. The dependence of the revival structure on the initial position, energy, and angular momentum of the wave packet and the connection to the energy spectrum is discussed in detail. The results are also compared to two other highly symmetrical 2D infinite well geometries with exact revivals, namely the square and equilateral triangle billiards. We also show explicitly how the classical periodicity for closed orbits in a circular billiard arises from the energy eigenvalue spectrum, using a WKB analysis.

quant-ph↗

arXiv2003

Comparing classical and quantum probability distributions for an asymmetric infinite well

We compare the classical and quantum mechanical position-space probability densities for a particle in an asymmetric infinite well. In an idealized system with a discontinuous step in the middle of the well, the classical and quantum probability distributions agree fairly well, even for relatively small quantum numbers, except for anomalous cases which are due to the unphysical nature of the potential. We are able to derive upper and lower bounds on the differences between the quantum and classical results. We also qualitatively discuss the momentum-space probability densities for this system using intuitive ideas about how much time a classical particle spends in various parts of the well. This system provides an excellent example of a non-trivial, but tractable, quantum mechanical bound state problem where the correlations between the amplitude and curvature of quantum mechanical wavefunctions can be easily compared to classical intuition about particle motion, with quantitative success, but also warning of possible surprises in non-physical limiting cases.

quant-ph↗

arXiv2003

Anatomy of a quantum `bounce'

We discuss some of the properties of the `collision' of a quantum mechanical wave packet with an infinitely high potential barrier, focusing on novel aspects such as the detailed time-dependence of the momentum-space probability density and the time variation of the uncertainty principle product $Δx_t \cdot Δp_t$. We make explicit use of Gaussian-like wave packets in the analysis, but also comment on other general forms.

quant-ph↗

arXiv1999

Extra-dimensional gravity and dijet production at gamma gamma colliders

In this note, we consider dijet production at gamma gamma colliders as a probe of recently proposed, large extra-dimensional gravity models. The exchange of virtual, spin-2 graviton towers (Kaluza-Klein excitations) significantly modifies the cross section, as compared to the Standard Model predictions. We find that, in order to maximize the value of the effective scale that can be probed at a given center-of-mass energy, a very severe p_T cut should be applied; in general, a p_T equal to approximately 46% of the e^+ e^- beam energy gives the highest reach. We find that we can probe the effective mass scale from about 2.7 TeV to 11.1 TeV, depending on the center-of-mass energy and assumptions about the model.

hep-ph↗

arXiv1998

Radiation zeros and scalar particles beyond the standard model

Standard radiation zeros arise from the factorization properties of tree-level amplitudes involving a massless photon and can occur when all charged particles in the initial and final state have the same sign. We investigate how several different processes involving new scalar particles beyond the standard model may exhibit radiation zeros and how this structure might be exploited to probe their electromagnetic structure. We focus on (i) unnoticed aspects of angular zeros in the process e- + e- --> Delta-- + gamma for doubly charged Higgs boson (or any bilepton) production and (ii) the process gamma + e- --> q + S/V for scalar (S) or vector (V) leptoquarks (LQs). We also discuss how factorized amplitudes and radiation zeros may appear in the gauge boson fusion production of non-conjugate leptoquark pairs via gamma + W --> S_i + S_j* in high energy ee reactions and how the zeros affect the production cross-sections for various types of scalar leptoquarks.

hep-ph↗

arXiv1998

Eliminating the low-mass axigluon window

Using recent collider data, especially on the hadronic width the Z0, we exclude axigluons in the currently allowed low-mass window, namely axigluons in the mass range 50 GeV < M_A < 120 GeV. Combined with hadron collider data from di-jet production, axigluons with masses below roughly 1 TeV are now completely excluded.

hep-ph↗

arXiv1997

Third-generation leptoquark decays and collider searches

Collider searches for first-, second-, and third-generation scalar (S) or vector (V) leptoquarks (LQs) focus on the quark-lepton decay modes S,V -> q l. For SU(2)-doublet and -triplet leptoquarks with a sufficiently large splitting between the components, decays involving real W-boson emission (such as S_2^{(+5/3)} -> S_2^{(+2/3)} W^{+} and others) become possible and can change the patterns of leptoquark decays. For third-generation leptoquarks, where these mass splittings might be large, such modes could dominate certain leptoquark decays as they are (if kinematically allowed) guaranteed to be of order g^2 where g is the electroweak coupling. We calculate the decay rates for all such processes involving SU(2)-doublet and triplet, scalar and vector leptoquarks. Standard limits on mass splittings from precision electroweak measurements imply that only such decays involving SU(2)-doublet scalar LQs are likely kinematically possible.

hep-ph↗

arXiv1997

Leptoquark production in ultrahigh-energy neutrino interactions revisited

The prospects for producing leptoquarks (LQs) in ultrahigh-energy (UHE) neutrino nucleon collisions are re-examined in the light of recent interpretations of HERA data in terms of leptoquark production. We update predictions for cross-sections for the production of first- and second-generation leptoquarks in UHE nu-N and nubar-N collisions including (i) recent experimental limits on masses and couplings from the LEP and TEVATRON colliders as well as rare processes, (ii) modern parton distributions, and (iii) radiative corrections to single leptoquark production. If the HERA events are due to an SU(2) doublet leptoquark which couples mainly to (e+,q) states, we argue that there are likely other LQ states which couple to neutrinos which are close in mass, due to constraints from precision electroweak measurements.

hep-ph↗

arXiv1997

Closing the window on the axigluon mass using top quark production data

The contribution of axigluons (the massive color-octet gauge bosons in all chiral color models) to top quark pair production in hadronic collisions is considered. The agreement between the experimental values of the ttbar production cross-section at the TEVATRON and recent QCD predictions is used to discuss limits on the axigluon mass. Specifically,intermediate mass axigluons, those in the mass range 50 GeV < M_A < 120 GeV which has not already been excluded, would increase the tree-level qqbar -> ttbar cross-section by a factor of >= 2, thereby increasing the theoretical predictions for sigma_{ttbar} by Delta sigma_{ttbar} = 3.2-3.7 pb (2.7-3.1 pb) using leading-order (next-to-leading order) parton distributions over this mass range, independent of the axigluon decay width. Such an increase is roughly 1.3-1.6 (0.9-1.2) standard deviations larger than that suggested by the apparent good agreement between combined experimental results and recent theoretical calculations and so is not ruled out, but is definitely disfavored. Future high-statistics top-quark production runs will likely make a more definitive statement. The forward-backward asymmetry in ttbar production induced by axigluons in this mass window is also discussed and found to be quite large and so could provide another constraint.

hep-ph↗

arXiv1995

Prospects for spin physics at RHIC

The proposal to perform polarized proton-proton collisions at collider energies at RHIC is reviewed. After a brief reminder of the desirability of high energy spin physics measurements, we discuss the machine parameters and detector features which are taken to define a program of spin physics at RHIC. Some of the many physics processes which can provide information on polarized parton distributions and the spin-dependence of QCD and the electroweak model at RHIC energies are discussed. (To appear in the proceedings of the International Symposium on Particle Theory and Phenomenology, Iowa State University, May 22-24, 1995.)

hep-ph↗

arXiv1994

Aspects of four-jet production in polarized proton-proton collisions

We examine the intrinsic spin-dependence of the dominant $gg \rightarrow gggg$ subprocess contribution to four-jet production in polarized proton-proton collisions using helicity amplitude techniques. We find that the partonic level, longitudinal spin-spin asymmetry, $\hat{a}_{LL}$, is intrinsically large in the kinematic regions probed in experiments detecting four isolated jets. Such events may provide another qualitative or semi-quantitative test of the spin-structure of QCD in planned polarized $pp$ collisions at RHIC.

hep-ph↗