Wave packet studies of nonadiabatic effects

Based on Kleinert's variational perturbation (KP) theory [Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed. (World Scientific, Singapore, 2004)], we present an analytic path-integral approach for computing the effective centroid potential. The approach enables the KP theory to be applied to any realistic systems beyond the first-order perturbation (i.e., the original Feynman-Kleinert [Phys. Rev. A 34, 5080 (1986)] variational method). Accurate values are obtained for several systems in which exact quantum results are known. Furthermore, the computed kinetic isotope effects for a series of proton transfer reactions, in which the potential energy surfaces are evaluated by density-functional theory, are in good accordance with experiments. We hope that our method could be used by non-path-integral experts or experimentalists as a "black box" for any given system.

The complex geometrical optics (CGO) was applied for the spatiotemporal evolution of arbitrary number of 3D mutually incoherent (with different carrier frequencies) Gaussian wave packets (GWPs) interacting and propagating in a nonlinear medium of Kerr type. The CGO reduced description of the propagation of the beam, the pulse and the wave packet to complex ordinary differential equations (ODE) This leads to exceptionally fast numerical algorithms. We observed high efficiency of the CGO method to compute interactions of arbitrary number of 3D Gaussian wave packets propagating in a nonlinear (anomalous) dispersive medium of the Kerr type. The derived CGO equations were compared with equations obtained by the variational method. CGO described the Gaussian beam propagation in free space as well as the Gaussian pulse spreading in the linear anomalous dispersive medium more illustratively than both the Fourier transform method and the Fresnel diffraction integral method. The spatiotemporal CGO has been proven to be a method more practical than the spectral analysis, the variational method, the method of moments and the method of the generalized eikonal approximation. Complementary to the presented results, an on-line CGO solver, implemented in Javascript, is freely available at the authors' website: http://slawek.ps.pl/odelia.html.

The propagation of finite amplitude baroclinic wave packets in the two-layer model is investigated by using the multiple-scale method. It is shown that the propagation of the wave packets can be described by the so-called unstable nonlinear Schrodinger equation which possesses envelope soliton solutions. The speeds of the solitons are independent of their amplitudes.while the width of the solitons is directly proportional to their speeds but inversely proportional to their amplitudes.

In this chapter, we discuss another aspect of the wave packet theory, wave packet progagation. Earlier chapters mainly focused on the structure and structural change of wave packets, without explicit discussions of the propagation property of wave packets, which is possible only when the structure independence theorem discussed in Chapter 2 holds. However, when the structure independence theorem is not valid, we have to take the propagation property of the wave packet into account, as discussed in Chapter 2. Moreover, the propagation property of wave packet is also of great importance. As shown in Chapter 2, the wave packet is always propagated along the group velocity, that is, the wave packet velocity. Many modern ideas on wave propagation were originated by Rayleigh, including the distinction between the phase velocity and the group velocity; this appears very early in Rayleigh’s work (1877). Since then, there have been numerous studies on wave packet propagation associated with the group velocity, including several monographs (Brillouin, 1946, 1953, 1960; Tolstoy, 1973). The study of wave packet propagation was started as soon as the concept of wave group or wave packet was introduced in wave mechanics (Brillouin, 1946).

Methods for the approximate numerical integration of the time dependent Schrödinger equation with given initial conditions (the initial wave packet) are presented. The methods are based on the Schrödinger representation of the quantum dynamic system. The quantum dynamic equations are transformed into Hamilton‐Jacobi type equations of motion as they occur in multi particle classical dynamics, i.e. standard techniques in molecular dynamics can be applied for the integration. The dynamics of minimum uncertainty Gaussian wave packets in strongly non‐harmonic, nonlinearly coupled oscillators are studied as examples. The numerically exact solutions are compared to time dependent SCF approximations of the wave packet.

Monthly or seasonally mean anomalies of large-scale atmospheric circulation are better represented by wave packets or their combination. Both qualitative and quantitative analyses of equations of wave packet dynamics, which are obtained by the use of WKB approximation, are very helpful for the understanding of structure, formation and propagation of stationary and quasi-stationary planetary wave packet patterns in the atmosphere. Indeed, these equations of wave packet dynamics can be directly solved by the method of characteristic lines, and the results can be simply and clearly interpreted by physical laws. In this paper, a quasi-geostrophic barotropic model is taken for simplicity, and the wave packets superimposed on several ideal profiles of the basic current and excited by some ideal forcings are investigated in order to make comparison of the accuracy of calculation with the analytical solution. It is revealed that (a) the rays of stationary planetary wave packet do not coincide with but go away from the great circle with significant difference if the shear of the basic zonal flow is not too small; (b) being superimposed on a westerly jet flow with positive shear (∂U λ/∂y>0), the stationary wave packets excited by low-latitudinal forcing are first intensified during their northeastward propagation in the Northern Hemisphere, then reach their maximum of amplitude at some critical latitude, and after that weaken again; (c) the connected line of extremes (the positive and negative centres) of wave packet does not coincide with but crosses the ray by an angle, the larger the scale of external forcing, the larger the angle; and (d) the whole pattern of a trapped stationary wave packet is complicated by the interference between the incident and reflected waves.

Effects of the medium fractionality and oscillating potential profiles on the Superarrivals of the Gaussian wave packets

We derive an analytical solution to the time-dependent Schrödinger equation for the transmission of a Gaussian wave packet through an arbitrary potential of finite range. We consider the situation where the initial Gaussian wave packet is sufficiently broad in momentum space to include the full resonance structure of the system in the dynamical description. We find that the transmitted wave packet may be written as the free evolving Gaussian wave packet solution times a transient term. We demonstrate that both at very large distances and very long times the transient term tends to the transmission amplitude of the system and hence the time-evolving solution reproduces the resonance spectra of the system. We also prove that at a fixed distance and very long times the analytical solution is t−3/2 which extends previous analysis on this issue to arbitrary finite-range potentials. Our results are exemplified for a multibarrier system.

Experimental studies are reported of oscillations and radiation that is spontaneously excited by an electron beam which is shot along a diverging magnetic field into a plasma from a hot cathode. In the present study we focus on oscillations below the electron gyrofrequency, where we find that whistler mode radiation appears in the form of bursts, or wave packets, each with typically 0.1–1 μs time duration, and which together cover typically a few percent of the full time. Wave packets are found in a broad frequency range of 7–40 MHz, while each individual wave packet is dominated by a single frequency. There is propagation along two routes: at the group velocity resonance cone angle, away from the central channel where the waves are excited, and in a channel along the magnetic field. Features of the whistler mode wave packets that are studied include (1) the statistics of amplitudes, frequencies, and time durations; (2) the propagation and decay of wave packets with different frequencies; (3) the group and phase velocities; and (4) how the wave packet production varies with the energy, and the current density, in the electron beam.

The generation and dynamics of plasmon wave packets in single gold nanorods were observed at a spatiotemporal scale of 100 nm and 10 fs via time-resolved near-field optical microscopy. Following simultaneous excitation of two plasmon modes of a nanorod with an ultrashort near-field pulse, a decay and revival feature of the time-resolved signal was obtained, which reflected the reciprocating motion of the wave packet. The time-resolved near-field images were also indicative of the wave packet motion. At some period of time after the excitation, the spatial features of the two modes appeared alternately, showing motion of plasmonic wave crests along the rod. The wave packet propagation was clearly demonstrated from this observation with the aid of a simulation model. The present experimental scheme opens the door to coherent control of plasmon-induced optical fields in a nanometer spatial scale and femtosecond temporal scale.

We discuss new aspects of generating and observing electronic wave packets in Rydberg systems. A new scheme of observing wave packets in a pump and probe configuration with time-delayed correlation broad-bandwidth laser pulses is proposed and wave packet generation in near threshold multiphoton ionization is discussed.

Ground-state energy of a polaron in the strong-coupling region is calculated. Variational method and second-order perturbation theory are used to improve the usual strong-coupling theory. The results obtained are compared with those by the Feynman theory.

Quantum state-selected dynamics of C((3)P) + OH (X(2)Π) → CO(a(3)Π) + H ((2)S) reaction on its first excited electronic potential energy surface (1(2)A(")) is examined here using a time-dependent wave packet propagation approach. All partial wave contributions for the total angular momentum, J = 0-95, are included to obtain the converged cross sections and initial state-selected rate constants in the temperature range of 10-500 K. The reaction probability, as a function of collision energy, exhibits dense oscillatory structures owing to the formation of resonances during collision. These resonance structures also persist in reaction cross sections. The effect of reagent rotational and vibrational excitation on the dynamical attributes is examined and discussed. Reagent rotational excitation decreases the reactivity whereas, vibrational excitation of the reagent has minor effects on the reactivity. The results presented here are in good accord with those obtained using the time-independent quantum mechanical and quasi-classical trajectory methods.

Gaussian wave packets (GWPs) are well suited as basis functions to describe the time evolution of arbitrary wave functions in systems with non-singular smooth potentials. They are rare in atomic systems on account of the singular behaviour of the Coulomb potential.We present a time-dependent variational method that makes the use of GWPs possible in the description of propagation of quantum states also in these systems. We use a regularization of the Coulomb potential and introduce a fictitious-time coordinate in which the evolution of an initial state can be calculated exactly and analytically for a pure Coulomb potential. Therefore, in perturbed atomic systems variational approximations only arise from those parts of the potentials which deviate from the Coulomb potential. The method is applied to the hydrogen atom in external magnetic and electric fields. It can be adapted to systems with definite symmetries and thus allows for a wide range of applications.

Total cross sections for electron scattering on “quasi spherical” (CH4, SiH4, GeH4) molecules have been analyzed phenomenologically over a wide energy range. Regions, at low and high energies can be usefully represented by simple analytical formulae. Regularities associated with characteristic points such as the Ramsauer minima have been exposed. Comparison with other simple hydrides (NH3, H2O, H2S) allows the demonstration of a possible correlation between the maximum value of the total cross section and the bond length. Some points of contact with first-principles theory are noted and in particular the energy at which the maximum cross section occurs, is related to the occurrence of a partial wave resonance. In the absence of complete data for GeH4, prediction of characteristic points in the low energy cross section proves possible via the phenomenological analysis. Similarly, in the high energy regime, predictions of the cross section for SnH4 is made from data on the lighter molecules of the series, using non-relativistic Thomas-Fermi self consistent field scaling.