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=9μa²/4?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°-60°-60°) triangle, such as the half equilateral (30°-60°-90°) triangle and other “foldings,” which have related energy spectra and revival structures.     

Metal interface formation studied by high-energy reflection energy loss spectroscopy and electron Rutherford backscattering

We demonstrate that high-energy, high-resolution reflection electron energy loss spectroscopy can provide unique insights into interface formation, especially for the case where an extended interface is formed. By changing the geometry and/or electron energy the electronic structure can be probed over a range of thicknesses (from 10s of Å to more than 1000 Å). At the same time one resolves the elastically scattered electrons into different components, corresponding to scattering of atoms with different mass (so-called ‘electron Rutherford backscattering’). Thus these high-energy REELS/elastic scattering experiments obtain information on both the electronic structure and the atomic composition of the overlayer formed.     

Quantum dynamics of hydrogen atom in complex space

We show in this paper that the electron’s quantum dynamics in hydrogen atom can be modeled exactly by quantum Hamilton-Jacobi formalism. It is found that the quantizations of energy, angular momentum, and the action variable ∫p dq are all originated from the electron’s complex motion, and that the shell structure observed in hydrogen atom is indeed originated from the structure of the complex quantum potential, from which the quantum forces acting upon the electron can be uniquely determined, the stability of atomic configuration can be justified, and the electron’s complex trajectories can be derived accordingly. Based on the derived electron’s trajectory, we can explain why the electron appears at some positions with large probability, while at some other positions with small probability. The positions with maximum probability predicted by standard quantum mechanics are found to be just the stable equilibrium points of the electron’s non-linear complex dynamics. The electron’s trajectories in hydrogen atom are discovered to be very diverse and strongly state-dependent; some of them are open and non-periodic, while some are closed and periodic. Over such a great diversity of orbits, commensurability condition ensuring the existence of closed orbit will be derived and the de Broglie’s standing wave pattern will be identified. Along the investigation of the electron’s orbits in hydrogen atom, we will also clarify why old quantum mechanics using the concept of classical orbit can correctly predict the energy quantization of hydrogen atom and meanwhile why it is not applicable to general quantum system. Finally, the internal mechanism of how the precessing, non-conical eigen-trajectories can evolve continuously to the classical, non-precessing, conical orbits as n → ∞ is explained in detail.     

Non-Abelian hydrodynamics and the flow of spin in spin-orbit coupled substances

Motivated by the heavy ion collision experiments there is much activity in studying the hydrodynamical properties of non-Abelian (quark-gluon) plasmas. A major question is how to deal with color currents. Although not widely appreciated, quite similar issues arise in condensed matter physics in the context of the transport of spins in the presence of spin-orbit coupling. The key insight is that the Pauli Hamiltonian governing the leading relativistic corrections in condensed matter systems can be rewritten in a language of SU(2) covariant derivatives where the role of the non-Abelian gauge fields is taken by the physical electromagnetic fields: the Pauli system can be viewed as Yang-Mills quantum-mechanics in a ‘fixed frame’, and it can be viewed as an ‘analogous system’ for non-Abelian transport in the same spirit as Volovik’s identification of the He superfluids as analogies for quantum fields in curved space time. We take a similar perspective as Jackiw and coworkers in their recent study of non-Abelian hydrodynamics, twisting the interpretation into the ‘fixed frame’ context, to find out what this means for spin transport in condensed matter systems. We present an extension of Jackiw’s scheme: non-Abelian hydrodynamical currents can be factored in a ‘non-coherent’ classical part, and a coherent part requiring macroscopic non-Abelian quantum entanglement. Hereby it becomes particularly manifest that non-Abelian fluid flow is a much richer affair than familiar hydrodynamics, and this permits us to classify the various spin transport phenomena in condensed matter physics in an unifying framework. The “particle based hydrodynamics” of Jackiw et al. is recognized as the high temperature spin transport associated with semiconductor spintronics. In this context the absence of faithful hydrodynamics is well known, but in our formulation it is directly associated with the fact that the covariant conservation of non-Abelian currents turns into a disastrous non-conservation of the incoherent spin currents of the high temperature limit. We analyze the quantum-mechanical single particle currents of relevance to mesoscopic transport with as highlight the Ahronov-Casher effect, where we demonstrate that the intricacies of the non-Abelian transport render this effect to be much more fragile than its abelian analog, the Ahronov-Bohm effect. We subsequently focus on spin flows protected by order parameters. At present there is much interest in multiferroics where non-collinear magnetic order triggers macroscopic electric polarization via the spin-orbit coupling. We identify this to be a peculiarity of coherent non-Abelian hydrodynamics: although there is no net particle transport, the spin entanglement is transported in these magnets and the coherent spin ‘super’ current in turn translates into electric fields with the bonus that due to the requirement of single valuedness of the magnetic order parameter a true hydrodynamics is restored. Finally, ‘fixed-frame’ coherent non-Abelian transport comes to its full glory in spin-orbit coupled ‘spin superfluids’, and we demonstrate a new effect: the trapping of electrical line charge being a fixed frame, non-Abelian analog of the familiar magnetic flux trapping by normal superconductors. The only known physical examples of such spin superfluids are the ³He A- and B-phase where unfortunately the spin-orbit coupling is so weak that it appears impossible to observe these effects.     

A theory of cosmic rays

We present a theory of non-solar cosmic rays (CRs) in which the bulk of their observed flux is due to a single type of CR source at all energies. The total luminosity of the Galaxy, the broken power-law spectra with their observed slopes, the position of the ‘knee(s)’ and ‘ankle’, and the CR composition and its variation with energy are all predicted in terms of very simple and completely ‘standard’ physics. The source of CRs is extremely ‘economical’: it has only one parameter to be fitted to the ensemble of all of the mentioned data. All other inputs are ‘priors’, that is, theoretical or observational items of information independent of the properties of the source of CRs, and chosen to lie in their pre-established ranges. The theory is part of a ‘unified view of high-energy astrophysics’ — based on the ‘Cannonball’ model of the relativistic ejecta of accreting black holes and neutron stars. The model has been extremely successful in predicting all the novel properties of Gamma Ray Bursts recently observed with the help of the Swift satellite. If correct, this model is only lacking a satisfactory theoretical understanding of the ‘cannon’ that emits the cannonballs in catastrophic processes of accretion.     

Switching an electrical current with atoms: the reproducible operation of a multi-atom relay

The demonstration of a multi-atom quantum point contact relay is reported, which can be reversibly switched between a quantized conducting ‘on-state’ and an insulating ‘off-state’ by applying an electrochemical control potential to a separate, third electrode, the control or gate electrode. The transition occurs directly from the conducting ‘on-state’ at 5 G₀ (G₀=2e²/h being the conductance quantum) to the insulating ‘off-state’. No stable intermediate levels are observed during the switching process, indicating a reproducible bistable reconfiguration of one single multi-atom contact rather than a deposition and dissolution of different parallel contacts. The results at the same time demonstrate the feasibility and reproducible operation of a configurable electronic device based on a multi-atom contact, which exhibits the functionality of an atomic relay or a transistor, opening intriguing perspectives for electronics and logics on the atomic scale.     

Detection of collective motions in dielectric spectra and the meaning of the generalized Vogel-Fulcher-Tamman equation

Based on the reduction property of dielectric spectra associated with the power-law function [∼(jωτ)^±ν] that appears in the frequency domain, one can develop an effective procedure for detection of different reduced motions (described by the corresponding power-law exponents) in temperature domain. If the power-law exponent ν is related to characteristic relaxation time τ by the relationship ν=ν₀ ln(τ/τ_s)/ln(τ/τ₀) (here τ_s, τ₀ are the characteristic times characterizing a movement over fractal cluster that is defined in Ref. [Ya.E. Ryabov, Yu. Feldman, J. Chem. Phys. 116 (2002) 8610]) and the simple temperature dependence of τ(T)=τ_A exp(E/T) obeys the traditional Arrhenius relationship, then one can prove that any extreme point figuring in the complex permittivity ε(jω) spectra (characterized by the values [ω_m, y(ω_m)]) obeys the generalized Vogel-Fulcher-Tamman (VFT) equation. This important statement confirms the existence of the ‘universal’ response (UR) (discovered and classified by Jonscher in frequency domain) and opens new possibilities in the detection of the ‘hidden’ collective motions in temperature region for self-similar (heterogeneous) systems. It gives also the extended interpretation of the VFT equation and allows one to differentiate collective motions passing through an extreme point. This differentiation, in turn, allows one to select the proper fitting function containing one or two (at least) relaxation times for the fitting of the complex permittivity function ε(jω) in the limited frequency domain. This conclusion can allow for the classification of dielectric spectroscopy as the spectroscopy of the reduced (collective) motions, which are described by different power-law exponents on the mesoscale region. The verification of this approach on available DS data (poly(ethylene glycol)-based-single-ion conductors) completely confirms the basic statements of this theory and opens new possibilities in general classification of different motions that can be detected in the analysis of the different dielectric permittivity spectra.     

Entwined pairs and Schrödinger’s equation

We show that a point particle moving in space-time on entwined-pair paths generates Schrödinger’s equation in a static potential in the appropriate continuum limit. This provides a new realist context for the Schrödinger equation within the domain of classical stochastic processes. It also suggests that ‘self-quantizing’ systems may provide considerable insight into conventional quantum mechanics.     

Scattering times in AlGaN/GaN two-dimensional electron gas from magnetotransport measurements

The various scattering times of two-dimensional electron gas were investigated in modulation-doped Al_0.22Ga_0.78N/GaN quantum wells by means of magnetotransport measurements. The ratio of transport and quantum scattering times, τ_t/τ_q∼1, shows that the dominant mobility-limiting mechanisms are short-range scattering potentials. The low-field magnetoresistance shows the weak antilocalization and localization phenomenon from which the spin-orbit scattering and inelastic scattering times are obtained. The inelastic scattering time is found to follow the T⁻¹ law, indicating that electron-electron scattering with small energy transfer is the dominant inelastic process.     

Thermodynamics of relation-based systems with applications in econophysics,sociophysics, and music

A methodology was developed to analyze relation-based systems evolving in time by using the fundamental concepts of thermodynamics. The behavior of such systems can be tracked from the scattering matrix which is actually a network of directed vectors (or pathways) connecting subsequent values, which characterize an event, such as the index values in stock markets. A system behaves in a rigid (elastic) way to an external effect and resists permanent deformation, or it behaves in a viscous (or soft) way and deforms in an irreversible way. It was shown in the past that a formula derived using the slope of paths gives a measure about the extent of viscoelastic behavior of relation-based systems Gündüz (2009) [5] Gündüz and Gündüz (2010) [6]. In this research the ‘work’ associated with ‘elastic’ component, and ‘heat’ associated with ‘viscous’ component were discussed and elaborated. In a simple two subsequent pathway system in a scattering diagram the first vector represents ‘the cause’ and the second ‘the effect’. By using work and heat energy relations that involve force and also storage and loss modulus terms, respectively, one can calculate the energy involved in relation-based systems. The modulus values can be found from the parallel and vertical components of the second vector with respect to the first vector. Once work-like and heat-like terms were determined the internal energy is also easily found from their summation. The parallel and vertical components can also be used to calculate the magnitude of torque and torque energy in the system. Three cases, (i) the behavior of the NASDAQ-100 index, (ii) a social revolt, and (iii) the structure of a melody were analyzed for their ‘work-like’, ‘heat-like’, and ‘torque-like’ energies in the course of their evolution. NASDAQ-100 exhibits highly dissipative behavior, and its work terms are very small but heat terms are of large magnitude. Its internal energy highly fluctuates in time. In the social revolt studied work and heat terms are of comparable magnitude. The melody depicts highly organized structure, and usually has larger work terms than heat terms, but at some intervals heat terms burst out and attain very large magnitudes. Torque terms reach high values when the system is recovering from a minimum value.     

矩形弹子球中的量子波包分析(英文)

利用波包分析量子力学体系的动力学行为在研究经典和量子的对应关系方面越来越成为一个非常重要的方法.利用高斯波包分析方法,我们计算了矩形弹子球体系的自关联函数,自关联函数的峰和经典周期轨道的周期符合的很好,这表明经典周期轨道的周期可以通过含时的量子波包方法产生.我们还讨论了矩形弹子球的波包回归和波包的部分回归,计算结果表明在每一个回归时间,波包出现精确的回归.对于动量为零的波包,初始位置在弹子球内部的特殊对称点处,出现一些时间比较短的附加的回归.     

Wavefunctions for topological quantum registers

We present explicit wavefunctions for quasi-hole excitations over a variety of non-abelian quantum Hall states: the Read-Rezayi states with k ? 3 clustering properties and a paired spin-singlet quantum Hall state. Quasi-holes over these states constitute a topological quantum register, which can be addressed by braiding quasi-holes. We obtain the braid properties by direct inspection of the quasi-hole wavefunctions. We establish that the braid properties for the paired spin-singlet state are those of ‘Fibonacci anyons’, and thus suitable for universal quantum computation. Our derivations in this paper rely on explicit computations in the parafermionic conformal field theories that underly these particular quantum Hall states.     

Quantization of contact manifolds and thermodynamics

The physical variables of classical thermodynamics occur in conjugate pairs such as pressure/volume, entropy/temperature, chemical potential/particle number. Nevertheless, and unlike in classical mechanics, there are an odd number of such thermodynamic co-ordinates. We review the formulation of thermodynamics and geometrical optics in terms of contact geometry. The Lagrange bracket provides a generalization of canonical commutation relations. Then we explore the quantization of this algebra by analogy to the quantization of mechanics. The quantum contact algebra is associative, but the constant functions are not represented by multiples of the identity: a reflection of the classical fact that Lagrange brackets satisfy the Jacobi identity but not the Leibnitz identity for derivations. We verify that this ‘quantization’ describes correctly the passage from geometrical to wave optics as well. As an example, we work out the quantum contact geometry of odd-dimensional spheres.     

On the phase-space analysis of photon mediated quantum state transfer in nanophotonic waveguidance

A cavity quantum electrodynamics (QED) based approach for transferring quantum state between quantum nodes has been proposed, wherein a rubidium (⁸⁷Rb) atom trapped inside a two-mode optical cavity forms the quantum node and photons serve as the information carrier between two such nodes. Information is encoded into polarized photon states generated through the application of a system of lasers. The focus is made on the phase-space analysis of the approach, wherein two subspaces of the hyperfine energy levels with magnetic sub-levels of rubidium (⁸⁷Rb) atom represent the logic states ‘0’ and ‘1’. The system of lasers initiates a cavity assisted Raman process which, in turn, generates a right- or left-circularly polarized photon depending on the logic state of the transmit node. Once the photon is received (at the receive node), the logic state of the transmit node is restored into the receive node through a cavity QED process.     

Thin spectrum states in bulk superconductors and superconducting grains

We show how a local pairing model for superconductivity can be used to describe the symmetry breaking mechanism in exact analogy to the cases of quantum crystals and antiferromagnets. We find that there are low energy states associated with the symmetry breaking process which are not influenced by the Anderson-Higgs mechanism. The presence of these ‘thin spectrum’ states in qubits based on superconducting material leads to a maximum time for which such qubits can remain quantum coherent. We also show how the charging energy of superconducting quantum dots may give the thin spectrum states a finite energy gap, impeding the spontaneous breaking of phase symmetry.     

Towards a novel simulation approach for the transport of atomic states through polarized photons

A novel approach for transferring logic states from one quantum node to other is proposed. Logic states ‘0’ and ‘1’ are represented by two subspaces of the hyperfine states space of rubidium atom (⁸⁷Rb). The atom, placed at the center of a two-mode cavity, is excited by simultaneous application of two laser beams, one for each subspace. Based on the logic state of the atom, it makes a transition to a higher energy level within the corresponding subspace. When the atom relaxes back to a lower state within the subspace, a left- or right-circularly polarized photon is emitted depending on whether the initial state was logic ‘0’ or logic ‘1’. The polarized photon leaks out of the cavity, reaches the receive node and gets detected therein. Simulation results show the efficacy of the approach.     

On the decay of148Pm

In the frame of a soluble coupled channels model we compare the exact lifetimeτ of a resonant state with two approximate estimates for thisτ. The exactτ value is related to the exact total widthΓ byτ=?Γ ^?1. The first approximate method has been derived by Fermi. It consists in a first order perturbation calculation ofΓ. The second one is Smith's time-delay method which leads to a direct relation betweenτ and the behaviour of theS-matrix along the real energy axis. We show that the Smith's method is the most useful one.