Semiclassical states in loop quantum gravity: an introduction

Unifying general relativity and quantum mechanics is a great challenge left to us by Einstein. To face this challenge, considerable progress has been made in non-perturbative canonical (loop) quantum gravity during the past 20 years. The kinematical Hilbert space of the quantum theory is constructed rigorously. However, the semiclassical analysis of the theory is still a crucial and open issue. In this review, we first introduce our work on constructing a semiclassical weave state, using the [ω] operator on the kinematical Hilbert space of loop quantum gravity. Then we give an introduction to the two different approaches currently investigated for constructing coherent states in the kinematical Hilbert space. The current status of semiclassical analysis in loop quantum gravity is then summarized.     

Semiclassical theory of spin-orbit interactions using spin coherent states

We formulate a semiclassical theory for systems with spin-orbit interactions. Using spin coherent states, we start from the path integral in an extended phase space, formulate the classical dynamics of the coupled orbital and spin degrees of freedom, and calculate the ingredients of Gutzwiller's trace formula for the density of states. For a two-dimensional quantum dot with a spin-orbit interaction of Rashba type, we obtain satisfactory agreement with fully quantum-mechanical calculations. The mode-conversion problem, which arose in an earlier semiclassical approach, has hereby been overcome.     

Semiclassical, trajectory-coherent, quasi-energy states for the one-dimensional schrödinger equation

On the basis of the complex WKB-Maslov method, semiclassical, trajectory-coherent, quasi-energy states are constructed for the one-dimensional Schrödinger equation with the potential corresponding to the well-known classical Duffing equation with a time-periodic driving force. Classical objects required to construct semiclassical quasi-energy states are determined by converging series in powers of a small parameter included in the potential. Leading terms of these series are calculated.     

Semiclassical dynamics of electron wave packet states with phase vortices

We consider semiclassical higher-order wave packet solutions of the Schr?dinger equation with phase vortices. The vortex line is aligned with the propagation direction, and the wave packet carries a well-defined orbital angular momentum (OAM) variant Planck's over 2pil (l is the vortex strength) along its main linear momentum. The probability current coils around the momentum in such OAM states of electrons. In an electric field, these states evolve like massless particles with spin l. The magnetic-monopole Berry curvature appears in momentum space, which results in a spin-orbit-type interaction and a Berry/Magnus transverse force acting on the wave packet. This brings about the OAM Hall effect. In a magnetic field, there is a Zeeman interaction, which, can lead to more complicated dynamics.     

To the theory of unstable states

Deviations of the decay law from exponents are discussing for a long time, however, experimental proofs of such deviations are absent. Here in the general form is shown that the conclusions about non-exponential contributions are due to the disregarding of advanced interactions, i.e. at principally non-relativistic considerations. We consider decay processes in the frame of interactions duration of the quantum field theory. We show that at this basis the usual exponential decay has place.     

Alpha-particle unstable states in12C

Alpha particle unstable states in¹²C have been investigated using the three reactions¹⁴N(d, α′)¹²C,¹⁴N(d, α′α)⁸Be and¹²C(α, α′)¹²C. Excitation cross sections and angular distributions have been measured for the reactions¹⁴N(d, α′) at 52 MeV and¹²C(α, α′) at 90 MeV. For a few angle pairs the α-decay of excited states in¹²C have been observed in a¹⁴N(d, α′α) correlation measurement. The reactions selectively excite onlyT=0 states. A previously undetected level with a large α-decay width (Γ=1.2 MeV) has been observed at 15.62 MeV excitation. This level shows up clearly in both reactions and is further distinguished from the nearby 14.08 state in the correlation measurement because of the distinctly different energy distributions of the decay products. On the basis of a particle angular distributions the 15.62 MeV level was assigned spin and parity 4⁺ and the level at 14.08 was assigned 3^?. The latter differs from the value suggest by earlier work. Comparison with DWBA calculations indicates that angular distributions of all other prominent levels are in agreement with their earlier assignments. Two levels at 19.20 and 20.30 MeV (both Γ?0.4 MeV) and three further levels at 21.81, 22.7 and 24.24 MeV also decay predominantly by α-emission.     

类氢及碱金属原子激发态寿命的半经典计算公式

根据对应原理,得到了类氢原子能态平均寿命半经典的计算公式τ(n,l),然后利用相对论单通道量子数亏损理论进行推广,得到用来计算考虑总角动量J的激发态寿命公式τ(n,l,l+1/2)和τ(n,l,l-1/2),利用单通道量子数亏损理论得到了碱金属原子n、l远大于1时激发态寿命的半经典公式τ(n.l)=τ0(m+M/nm/v/+M)2v7l(l+1/2)/n4,其计算结果和实验数据符合的很好.     

Stabilizing unstable steady states using extended time-delay autosynchronization

We describe a method for stabilizing unstable steady states in nonlinear dynamical systems using a form of extended time-delay autosynchronization. Specifically, stabilization is achieved by applying a feedback signal generated by high-pass-filtering in real time the dynamical state of the system to an accessible system parameter or variables. Our technique is easy to implement, does not require knowledge of the unstable steady state coordinates in phase space, automatically tracks changes in the system parameters, and is more robust to broadband noise than previous schemes. We demonstrate the controller's efficacy by stabilizing unstable steady states in an electronic circuit exhibiting low-dimensional temporal chaos. The simplicity and robustness of the scheme suggests that it is ideally suited for stabilizing unstable steady states in ultra-high-speed systems. (c) 1998 American Institute of Physics.     

Semiclassical approximation for relativistic potentials

We consider the application of semiclassical approximation to relativistic potentials for massless particles where the kinetic energy is a nontrivial, nonlocal operator. Quantization rules are derived for an arbitrary confining potential and compared to some exact results forS-waves. These results admit of a partial generalization to smalll values.     

Extracting and stabilizing the unstable states of hysteresis loops

A novel perturbation method for stabilizing the unstable intermediate states of hysteresis loops (i.e. S-shaped curves) is proposed. This method only needs output signals of the system to construct the perturbation form without delay-coordinate embedding technique, it is more practical for real-world systems. Stabilizing and tracking the unstable intermediate branches are demonstrated through the examples of a bistable laser system and a delay feedback system. All the numerical results are obtained by simulating each of the real experimential conditions.     

Control of unstable steady states in neutral time-delayed systems

We present an analysis of time-delayed feedback control used to stabilize an unstable steady state of a neutral delay differential equation. Stability of the controlled system is addressed by studying the eigenvalue spectrum of a corresponding characteristic equation with two time delays. An analytic expression for the stabilizing control strength is derived in terms of original system parameters and the time delay of the control. Theoretical and numerical results show that the interplay between thecontrol strength and two time delays provides a number of regions in the parameter space where the time-delayed feedback control can successfully stabilize an otherwise unstable steady state.     

Semiclassical theory for two-anyon system

The semiclassical quantization conditions for all partial waves are derived for bound states of two interacting anyons in the presence of a uniform background magnetic field. Singular Aharonov-Bohm type interactions between the anyons are dealt with by the modified WKB method of Friedrich and Trost. For s-wave bound state problems in which the choice of the boundary condition at short distance gives rise to an additional ambiguity, a suitable generalization of the latter method is required to develop a consistent WKB approach. We here show how the related self-adjoint extension parameter affects the semiclassical quantization condition for energy levels. For some simple cases admitting exact answers, we verify that our semiclassical formulas in fact provide highly accurate results over a broad quantum number range.     

Semiclassical quantum mechanics. III. The large order asymptotics and more general states

We solve the time dependent Schrödinger Equation $\text{i?(}\text{?ψ}\text{?t}\text{) = ?(?}{}^2\text{/2m) Δψ + Vψ}$ modulo errors which have L² norms on the order of $\text{?}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ for arbitrarily large l. The initial conditions we consider are fairly general states whose position and momentum uncertainties are proportional to $\text{?}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.