Tomographic Description of a Quantum Wave Packet in an Accelerated Frame

The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton’s law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.     

Features of the tunneling of a system with an internal degree of freedom through a potential barrier

The tunneling of a quantum system with an internal degree of freedom through a potential barrier is considered. Based on the exact numerical solution to the nonstationary Schrödinger equation, the tunneling of a model two-particle system through a potential barrier is studied and the dependences of the tunneling transparency of the barrier on the parameters of the wave packet that describes the system at the initial moment are obtained. A sharp increase in the tunneling probability related to the formation of a long-lived quasibound state of the system in the barrier region is demonstrated. A simple analytical model of the tunneling of a system with an internal degree of freedom that allows for a qualitative interpretation of the main features of the tunneling is constructed.     

Stationary phase method and delay times for relativistic and non-relativistic tunneling particles

The stationary phase method is frequently adopted for calculating tunneling phase times of analytically-continuous Gaussian or infinite-bandwidth step pulses which collide with a potential barrier. This report deals with the basic concepts on deducing transit times for quantum scattering: the stationary phase method and its relation with delay times for relativistic and non-relativistic tunneling particles. After reexamining the above-barrier diffusion problem, we notice that the applicability of this method is constrained by several subtleties in deriving the phase time that describes the localization of scattered wave packets. Using a recently developed procedure - multiple wave packet decomposition - for some specifical colliding configurations, we demonstrate that the analytical difficulties arising when the stationary phase method is applied for obtaining phase (traversal) times are all overcome. In this case, we also investigate the general relation between phase times and dwell times for quantum tunneling/scattering. Considering a symmetrical collision of two identical wave packets with an one-dimensional barrier, we demonstrate that these two distinct transit time definitions are explicitly connected. The traversal times are obtained for a symmetrized (two identical bosons) and an antisymmetrized (two identical fermions) quantum colliding configuration. Multiple wave packet decomposition shows us that the phase time (group delay) describes the exact position of the scattered particles and, in addition to the exact relation with the dwell time, leads to correct conceptual understanding of both transit time definitions. At last, we extend the non-relativistic formalism to the solutions for the tunneling zone of a one-dimensional electrostatic potential in the relativistic (Dirac to Klein-Gordon) wave equation where the incoming wave packet exhibits the possibility of being almost totally transmitted through the potential barrier. The conditions for the occurrence of accelerated and, eventually, superluminal tunneling transmission probabilities are all quantified and the problematic superluminal interpretation based on the non-relativistic tunneling dynamics is revisited. Lessons concerning the dynamics of relativistic tunneling and the mathematical structure of its solutions suggest revealing insights into mathematically analogous condensed-matter experiments using electrostatic barriers in single- and bi-layer graphene, for which the accelerated tunneling effect deserves a more careful investigation.     

Time dependence of resonant tunneling in heterostructures with a three-trough energy spectrum

The time dependence of resonant electron tunneling inGaAs andAlAs quantum heterostructures is studied for a three-trough model. From an analysis of transmitted and reflected wave phases, the spectra of tunneling and reflection times are obtained. The tunneling of a wave packet through a two-barrier heterostructure is modeled by numerical solution of the nonstationary Schrödinger equation.     

Dynamics of a wave packet in a system with a quantum dot in the Coulomb blockade regime

The nonstationary problem of electron tunneling through a quantum dot in the Coulomb block-ade regime is studied. The temporal Schrödinger equation is solved and the dynamics of the wave packet in a system consisting of a quantum dot connected to two one-dimensional contacts is investigated. The transmission coefficient is calculated. Dependences of the transmission on the tunneling electron energy are constructed.     

Probability representation of the quantum evolution and energy-level equations for optical tomograms

The von Neumann evolution equation for the density matrix and the Moyal equation for the Wigner function are mapped onto the evolution equation for the optical tomogram of the quantum state. The connection with the known evolution equation for the symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for optical tomograms. The classical Liouville equation for optical tomogram is obtained. An example of the parametric oscillator is considered in detail.     

CH4和CD4解离吸附的量子含时动力学研究

运用含时波包法(time-dependent wave packet method),对CH4和CD4在光滑静止的Ni(100)表面的解离吸附进行了量子动力学研究与计算.不同振动态下解离几率随平动能的变化曲线表明,反应分子的振动能对分子的解离有重要贡献,其反应趋势,与其它理论模型得到的结果一致.CH4与CD4解离几率的对数随平动能的变化曲线表明,CH4的解离几率比CD4的要高得多,这种同位素效应,是由它们不同的零点能和量子隧道效应引起的,且与实验结果符合得比较好.     

CH_4和CD_4解离吸附的量子含时动力学研究

运用含时波包法 (time -dependentwavepacketmethod) ,对CH4和CD4在光滑静止的Ni( 10 0 )表面的解离吸附进行了量子动力学研究与计算。不同振动态下解离几率随平动能的变化曲线表明 ,反应分子的振动能对分子的解离有重要贡献 ,其反应趋势 ,与其它理论模型得到的结果一致。CH4与CD4解离几率的对数随平动能的变化曲线表明 ,CH4的解离几率比CD4的要高得多 ,这种同位素效应 ,是由它们不同的零点能和量子隧道效应引起的 ,且与实验结果符合得比较好     

Time-dependent approaches for reaction and response in unstable nuclei

Time-dependent approaches for computation of quantum mechanical problems are presented for few-body models. Quantum tunneling in nuclear fusion is described using a time-dependent wave packet. Breakup and transfer reactions of ¹¹Be are visualized in the time-dependent manner. We also present a method to calculate strength functions using the time-dependent Schr?dinger equation. E1 strength distributions of halo nuclei, ¹¹Be and ⁶He, are presented.     

Quantile motion and tunneling

The concepts of quantile position, trajectory, and velocity are defined. For a tunneling quantum mechanical wave packet, it is proved that its quantile position always stays behind that of a free wave packet with the same initial parameters. In quantum mechanics the quantile trajectories are mathematically identical to Bohm's trajectories. A generalization to three dimensions is given.     

Transmission times of wave packets tunneling through barriers

The transmission of wave packets through barriers by tunneling is studied in detail by the method of quantum molecular dynamics. The distribution of the arrival times of a tunneling packet in front of and behind a barrier and the momentum distribution function of the packet are calculated. The average position and average momentum of the packet and their spread are investigated. It is found that below the barrier a part of the packet is reflected, and a Gaussian barrier increases the average momentum of the transmitted packet and its spread in momentum space. Zh. éksp. Teor. Fiz. 115, 1872–1889 (May 1999)     

Time and energy dependent dynamics of the STM tip — graphene system

Probability current and probability density of wave packets was calculated by solving the three dimensional time-dependent Schrödinger equation for a local potential model of the scanning tunneling microscope (STM) tip — graphene system. Geometrical and electronic structure effects of the three dimensional tunneling process are identified by studying three models of increasing complexity: a jellium half space, a narrow jellium sheet, and a local one electron pseudopotential. It was found that some of the key characteristics of the STM tip — graphene tunneling process are already present at the simple jellium models. In the STM tip — jellium half space system the direction of the momentum does not change during the tunneling event, hence this setup is characterised by introducing an effective distance. For the STM tip — narrow jellium sheet system the direction of the momentum is changed from vertical to horizontal during the tunneling event. The wave packet preferentially tunnels into the bound state of the jellium sheet. For the atomistic model of the graphene sheet an anisotropic spreading of the wave packet was found for hot electrons. This may open new opportunities to build carbon based nanoelectronic devices.     

Standard quantum mechanics featuring probabilities instead of wave functions

A new formulation of quantum mechanics (probability representation) is discussed. In this representation, a quantum state is described by a standard positive definite probability distribution (tomogram) rather than by a wave function. An unambiguous relation (analog of Radon transformation) between the density operator and a tomogram is constructed both for continuous coordinates and for spin variables. A novel feature of a state, tomographic entropy, is considered, and its connection with von Neumann entropy is discussed. A one-to-one map of quantum observables (Hermitian operators) on positive probability distributions is found.     

Relativistic quantum and classical kinetic equations in the tomographic-probability representation

Using the Radon integral transform of the relativistic kinetic equation for a spin-zero particle, we obtain the classical and quantum evolution equations for the tomographic probability density (tomogram) describing the states of the particle in both the classical and quantum pictures. The Green functions (propagators) of the evolution equations of a free particle are constructed. The examples of the evolution of Gaussian tomogram is considered.     

High resolution finite volume scheme for the quantum hydrodynamic equations

The theory of quantum fluid dynamics (QFD) helps nanotechnology engineers to understand the physical effect of quantum forces. Although the governing equations of quantum fluid dynamics and classical fluid mechanics have the same form, there are two numerical simulation problems must be solved in QFD. The first is that the quantum potential term becomes singular and causes a divergence in the numerical simulation when the probability density is very small and close to zero. The second is that the unitarity in the time evolution of the quantum wave packet is significant. Accurate numerical evaluations are critical to the simulations of the flow fields that are generated by various quantum fluid systems.A finite volume scheme is developed herein to solve the quantum hydrodynamic equations of motion, which significantly improve the accuracy and stability of this method. The QFD equation is numerically implemented within the Eulerian method. A third-order modified Osher–Chakravarthy (MOC) upwind-centered finite volume scheme was constructed for conservation law to evaluate the convective terms, and a second-order central finite volume scheme was used to map the quantum potential field. An explicit Runge–Kutta method is used to perform the time integration to achieve fast convergence of the proposed scheme.In order to meet the numerical result can conform to the physical phenomenon and avoid numerical divergence happening due to extremely low probability density, the minimum value setting of probability density must exceed zero and smaller than certain value. The optimal value was found in the proposed numerical approach to maintain a converging numerical simulation when the minimum probability density is 10^?5 to 10^?12. The normalization of the wave packet remains close to unity through a long numerical simulation and the deviations from 1.0 is about 10^?4.To check the QFD finite difference numerical computations, one- and two-dimensional particle motions were solved for an Eckart barrier and a downhill ramp barrier, respectively. The results were compared to the solution of the Schrödinger equation, using the same potentials, which was obtained using by a finite difference method. Finally, the new approach was applied to simulate a quantum nanojet system and offer more intact theory in quantum computational fluid dynamics.     

Probability of boundary conditions in quantum cosmology

One of the main interest in quantum cosmology is to determine boundary conditions for the wave function of the universe which can predict observational data of our universe. For this purpose, we solve the Wheeler–DeWitt equation for a closed universe with a scalar field numerically and evaluate probabilities for boundary conditions of the wave function of the universe. To impose boundary conditions of the wave function, we use exact solutions of the Wheeler–DeWitt equation with a constant scalar field potential. These exact solutions include wave functions with well known boundary condition proposals, the no-boundary proposal and the tunneling proposal. We specify the exact solutions by introducing two real parameters to discriminate boundary conditions, and obtain the probability for these parameters under the requirement of sufficient e-foldings of the inflation. The probability distribution of boundary conditions prefers the tunneling boundary condition to the no-boundary boundary condition. Furthermore, for large values of a model parameter related to the inflaton mass and the cosmological constant, the probability of boundary conditions selects an unique boundary condition different from the tunneling type.     

The Quantum-Classical Comparison of the Arrival-Time Distribution Through the Probability Current

We consider the arrival time distribution defined through the quantum probability current for a Gaussian wave packet representing free particles in quantum mechanics in order to explore the issue of the classical limit of arrival time. We formulate the classical analogue of the arrival time distribution for an ensemble of free particles represented by a phase space distribution function evolving under the classical Liouville's equation. The classical probability current so constructed matches with the quantum probability current in the limit of minimum uncertainty. Further, it is possible to show in general that smooth transitions from the quantum mechanical probability current and the mean arrival time to their respective classical values are obtained in the limit of large mass of the particles.