Narrow Results

Using numerical simulations of the time-dependent Schrödinger equation, we study the full quantum dynamics of the motion of an atomic ion in a linear Paul trap. Such a trap is based on a time-varying, periodic electric field and hence corresponds to a time-dependent potential for the ion, which we model exactly. We compare the center-of-mass motion with that obtained from classical equations of motion, as well as to results based on a time-independent effective potential. We also study the oscillations of the width of the ion’s wave packet, including close to the border between stable (bounded) and unstable (unbounded) trajectories. Our results confirm that the center-of-mass motion always follows the classical trajectory, that the width of the wave packet is bounded for trapping within the stability region, and therefore that the classical trapping criterion is fully applicable to quantum motion.

The operator expansion of free Green function of Helmholtz equation for arbitrary N-dimensional space leads to asymptotic extension of three dimensions Grimus–Stockinger formula closely related to multipole expansion. Analytical examples inspired by neutrino oscillation and neutrino deficit problems are considered for relevant class of wave packets.

In this paper, we study the origin of the quantum particle entanglement. Particles will have one mixed wave function as soon as they are created, which are called quantum particle entanglement. Electron spin states are used as an example to discuss this topic. When two electrons are created simultaneously, they have two different mixed quantum spin states. Before the measurement of its spin, we cannot determine its spin state. However, as soon as the spin of one of the electrons is determined (measured), the spin of the other will definitely be in the opposite state, regardless of how far they are away from each other. This paper uses the mechanism that the wave packet spreads as soon as they are created and then the wave packet shrinks when it undergoes a measurement to interpret this spooky phenomenon mentioned above.

In this note, we consider the mathematical aspects of the functional mechanics proposed and developed by I. V. Volovich. In particular, we briefly review the Liouville equation used to describe the dynamics of wave packets, study the characteristics of wave packets, such as the dynamics of their averages and dispersions, the features of dynamics in the case of Hamiltonian and Lagrangian theory, and give a number of simple examples illustrating general constructions.

In this paper, the analytic solution of the time-dependent Schrödinger equation is obtained for the wave packet in two-dimensional oscillator potential. The quantum dynamics of the wave packet is investigated based on this analytic solution. To our knowledge, this is the first time we solve, analytically and exactly this kind of time-dependent Schrödinger equation in a two-dimensional system, in which the Gaussian parameters satisfy the coupled nonlinear differential equations. The coherent states and their rotations of the system are discussed in detail. We find also that this analytic solution includes four kinds of modes of the evolutions for the wave packets: rigid, rotational, vibrational states and a combination of the rotation and vibration without spreading.

We consider wave packet propagation in mesoscopic quantum systems. A number of approaches are compared to look at the general solution of a time-dependent Schrödinger equation and the validity of the Ehrenfest theorem. Detailed calculations are presented to illustrate the results of a charged particle motion in the time-dependent systems, and show that the Ehrenfest theorem is not directly applicable in topologically nontrivial systems.

A treatment of the trembling motion or Zitterbewegung (ZB) phenomena in the armchair graphene nanoribbons (AGNRs) by using long-wave approximation is presented theoretically. We are first interested to study the time dependence of the average values for the current density in AGNR. The longitudinal and transversal components of the current density operator are derived analytically in the Heisenberg representation. The wave packet in a Gaussian distribution is considered with half-width d and a carrier wave vector in the longitudinal orientation k${}_{xo}$. The average values of the current density which represent the current induced by the motion of the electrons along the nanoribbon are calculated numerically. The interference between two energy branches, or the corresponding upper and lower energy states, leads to the trembling motion in the armchair graphene nanoribbon, and hence, our results emphasized that the phenomena of ZB has an aperiodic and nontransient oscillation in the longitudinal and transversal direction, respectively. The average values for the current density for AGNRs are calculated with extremely large value of N and compared with infinite pristine graphene.

A theory of the spatially localized light observed by Hau et al. [Nature 397 (1999) 594] is formulated on the basis of the Maxwell equations, where the dynamics of the electron currents due to the electromagnetic fields is also taken into account within the self-consistent linear response approximation. Formulae for the energy, momentum and angular momentum of the localized light are derived.

It is shown that the dissipative chaotic dynamics of a charged particle in the field of a wave packet with an arbitrary but finite number of harmonics can be reliably suppressed by judiciously varying the constant phase of the main harmonic, ϕ₀, while keeping null the corresponding constant phases of the remaining harmonics. The dependence of the chaotic threshold on the wave packet parameters is predicted theoretically (Melnikov method) and confirmed numerically (Lyapunov exponents). In particular, it is shown that ϕ₀ is effective at suppressing the chaotic behavior existing when ϕ₀ = 0 over a wide range of values of the wave packet width, while the remaining parameters are kept constant.

Classical point particles in Newtonian gravity obey, as they do in general relativity, the universality of free fall. However, classical structured particles, (for instance with a mass quadrupole moment), need not obey the universality of free fall. Quantum mechanically, an elementary “point” particle (in the particle physics sense) can be described by a localized wave packet, for which we can define a probability quadrupole moment. This probability quadrupole can, under plausible hypotheses, affect the universality of free fall. (So point-like elementary particles, in the particle physics sense, can and indeed must nevertheless have structure in the general relativistic sense once wave packet effects are included.) This raises an important issue of principle, as possible quantum violations of the universality of free fall would fundamentally impact on our ideas of what “quantum gravity” might look like. I will present an estimate of the size of the effect, and discuss where if at all it might be measured.

The ladder transitions controlled by two harmonic pulses are investigated theoretically using a time-dependent quantum wave packet method for the ground electronic state of HF molecule. By choosing $\omega +2\omega$, $\omega +3\omega$ and $\omega +4\omega$ schemes, the population can be transferred to target states $|\nu ,j〉$. The population distribution can be controlled by the average amplitude of total electric field which depends on the relative phase of two pulses. With the variation of the relative phase between $\omega$ and $3\omega$ pulses, the variation of population has a period of $2\pi$. For $\omega +2\omega$ and $\omega +4\omega$ schemes, the population distributions show oscillation behavior with a period of $\pi$ by varying the relative phase. The two harmonic pulses can realize a nearly complete population transfer to the target state.

The processes of population transfer in the ground electronic state of HCl molecule through the three transition schemes are investigated by numerically solving the time-dependent Schrödinger equation. Two harmonic pulses are employed to induce population transfer and the relative phase of the two pulses can control the final population distributions. In the ladder transition scheme, the variation range of the target population with the relative phase is nearly 100% which is larger than that in the multi-photon transition scheme. It is more efficient for the mixed transition scheme to control population transfer between the initial and target states by using the relative phase. Comparing with the multi-photon and ladder schemes, the transition probability of the target population is more sensitive to the two pulse amplitudes in the mixed transition scheme.

Sensing and perceiving involve enormous numbers of widely distributed dendritic and action potentials in cortex, before, during and after stimulus arrival but with differing spatiotemporal patterns. Stimulus-activated receptors drive cortical neurons directly (olfactory) or indirectly through thalamocortical relays. The driven activity induces hemisphere-wide, self-organized patterns of neural activity called wave packets. Three levels of brain function are hypothesized to mediate transition from sensation and perception. Microscopic activity expressed by action potentials is sensory. Macroscopic activity of the whole forebrain expressed by behavior is perceptual. Mesoscopic activity bridges the gap by the formation of wave packets. They form when sensory input destabilizes the primary receiving areas by local state transitions. The sensory-driven action potentials condense into mesoscopic wave packets like molecules forming raindrops from vapor. The condensation disks sustain 2D spatial patterns of phase and amplitude of carrier waves in the beta and gamma EEG. The AM patterns correlate not with features but with the context and value of sensory stimuli for the subjects, in a word, their meaning. The wave packets from all sensory areas are broadly transmitted through the forebrain. They induce the formation of macroscopic patterns that coalesce like scintillating pools over much and perhaps all of each hemisphere. The prediction is made for clinical testing that wave packets are precursor to states of awareness. They are not by themselves accessible to experience, as may be the macroscopic states initiated by global state transitions.

In this paper, we introduce the notion of finite wave packet groups for positive prime integers, as groups of cyclic dilations, translations and modulations. We will present a unified group theoretical approach for the cyclic wave packet transform on finite Abelian groups of prime order. Then, we study basic properties of this windowed transform.