A Multiple Scattering Theory Approach to Solving the Time-Dependent Schrödinger Equation with an Asymmetric Rectangular Potential

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.

Bohmian trajectories are considered for a particle that is free (i.e. the potential energy is zero), except for a half-line barrier. On the barrier, both Dirichlet and Neumann boundary conditions are considered. The half-line barrier yields one of the simplest cases of diffraction. Using the exact time-dependent propagator found by Schulman, the trajectories are computed numerically for different initial Gaussian wave packets. In particular, it is found that different boundary conditions may lead to qualitatively different sets of trajectories. In the Dirichlet case, the particles tend to be more strongly repelled. The case of an incoming plane wave is also considered. The corresponding Bohmian trajectories are compared with the trajectories of an oil drop hopping on the surface of a vibrating bath.

Using an analytical model potential which contains resonant and bound states, we show that the decay of the resonances can be simulated by Lindblad operators. For that purpose, the unitary time evolution of an initial Gaussian wave packet in the model potential is compared with the non-unitary time evolution, obtained by solving the Lindblad equation, of the same wave packet in a potential which coincides with the model potential in the region of interest but does not contain resonances. In the latter case, dissipative effects are accounted for by Lindblad operators which lead to phenomenological friction and diffusion constants in the equations of motion. We suggest how those constants can be determined in a non-heuristic way, being directly connected to the width of the resonance in the model potential which we calculate using the complex rotation method.

Time-independent wave-packet forms of the Schr\"odinger equation (TIWSE) and Lippmann-Schwinger equation (TIWLSB) have been derived by a partial time-to-energy Fourier transform of ${\mathit{L}}^{2}$ wave-packet solutions to the time-dependent Schr\"odinger equation. The new equations retain the initial wave packet \ensuremath{\chi}(${\mathit{t}}_{0}$) as a ``universal source'' of scattered waves, which applies for all collision energies E contained in the initial wave packet. The relationship between the solution ${\mathrm{\ensuremath{\Psi}}}_{\mathit{t}0}$(E) of the TIWSE or TIWLSE and the scattering solution ${\mathrm{\ensuremath{\Psi}}}^{(+)}$(E) of the standard time-independent Lippmann-Schwinger or Schr\"odinger equation is given and the method illustrated by a computational application.

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.

The molecular dynamics of Autler-Townes (AT) splitting in a multiphoton resonance ionization spectrum is numerically simulated using the time-dependent wave packet method for a rotating ${\mathrm{Na}}_{2}$ molecule with a linearly polarized laser pulse. Employing the Fourier grid method and Legendre polynomials as basis functions, the initial wave packet is directly propagated using the split operator method. Wave packet dynamics simulations demonstrate that the AT splitting results from rapid Rabi oscillations induced by an ultrashort intense laser pulse in rovibrational coherent states that belong to different electronic terms. But Rabi oscillations will be damped because of the effects of rotation and alignment of the molecule in laser fields. The AT splitting in a photoelectron spectrum could be observed for a well-aligned molecule with sufficiently rapid Rabi oscillations. Only an adequately short laser pulse can induce AT splitting in a photoelectron spectrum. The effect of initial rotational temperature on the molecular dynamics is also studied.

We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

Density functional theory (DFT) and multiple scattering theory (MST) have been used extensively for calculations in condensed matter physics of the electronic structure of materials built from atomic species throughout the periodic table. As originally formulated for the muffin-tin model, MST in general gave disappointing results in applications to molecules, which discouraged use of this methodology. Recent developments in so-called ‘full-potential’ MST have removed these muffin-tin constraints while retaining the computational efficiency associated with these methods. Full-potential MST is reviewed here, with emphasis on the theoretical basis of computational methods that are viable for molecules. Several recent developments in the formalism of DFT are reviewed, in particular reference-state DFT, which frees the theory from the constraint that the electronic density in the variational model state must be identical to that of the true ground state, and places DFT in the context of the Fermi liquid theory of Landau. The theory of self-interaction corrections is reexamined from this point of view.

Summary form only given. Wave packets (WPs) has recently emerged as a very active research field in quantum optics. We show here that the evolution of WPs may result in new class of large-scale interference effects: the formation of highly regular spatio-temporal patterns in the QM probability, |/spl Psi/|/sup 2/, with the key mechanism for this being pair interference between the eigen-modes of the system and the degeneracy of this interference. The new patterns have been discovered in the probability density |/spl Psi/|/sup 2/(x,t) of a particle in an infinitely deep square well, by numerical simulations of the evolution of a Gaussian WP; later work by a few groups used other initial conditions and media (e.g. Bose-Einstein condensate and angular WPs). In all these cases, the QM distribution shows a net of canals (minima), and ridges (maxima), running as straight lines in space-time (x,t). Similar patterns in the near field of a diffraction grating, and in matter waves in atom optics reveals a new aspects of the Talbot effect. So far the numerical exploration of these patterns has relied on the specific choice (i) of the initial conditions and (ii) of the square well; no physical explanation of the patterns has been suggested. Recently, we observed a quantum carpet under more general conditions, offered its universal explanation, and predicted distinct curved patterns in general case of an anharmonic potential.

Fewest-switches surface hopping (FSSH) is a popular nonadiabatic dynamics method which treats nuclei with classical mechanics and electrons with quantum mechanics. In order to simulate the motion of a wave packet as accurately as possible, standard FSSH requires a stochastic sampling of the trajectories over a distribution of initial conditions corresponding, e.g., to the Wigner distribution of the initial quantum wave packet. Although it is well-known that FSSH does not properly account for decoherence effects, there is some confusion in the literature about whether or not this averaging over a distribution of initial conditions can approximate some of the effects of decoherence. In this paper, we not only show that averaging over initial conditions does not generally account for decoherence, but also why it fails to do so. We also show how an apparent improvement in accuracy can be obtained for a fortuitous choice of model problems, even though this improvement is not possible, in general. For a basic set of one-dimensional and two-dimensional examples, we find significantly improved results using our recently introduced augmented FSSH algorithm.

Quantum revival is described as the time-periodic reconstruction of a wave packet initially localized in space and time. This effect is expected in finite-size systems which exhibit commensurable discrete spectrum such as the infinite quantum well. Here, we report on the experimental observation of full and fractional quantum revival for classical waves in a two dimensional cavity. We consider flexural waves propagating in thin plates, as their quadratic dispersion at low frequencies mimics the dispersion relation of quantum systems governed by Schrödinger equation. Time-dependent excitation and measurement are performed at ultrasonic frequencies and reveal a periodic reconstruction of the initial elastic wave packet.

We investigate the propagation of localized three-dimensional spatiotemporal Airy self-accelerating parabolic-cylinder light bullets in a linear medium. In particular, we consider the effects resulting from utilizing initial finite-energy Airy wave packets to accelerate these localized beams in the absence of any external potential. A general localized light bullet solution with the joint Airy pulse characteristics and parabolic-cylinder spatial characteristics is obtained in the Cartesian coordinates, using parabolic-cylinder and Airy functions. Our results show that the localized wave packets can retain their intensity features and still be accelerated over several Rayleigh lengths.

We are investigating the quantum dynamics of a well-collimated electron beam transmitting through planar channels of the Si crystal. Electron states were represented by wave packets while the electron beam is treated as an ensemble of noninteracting wave packets. We have investigated the relationship between classical caustic pattern and anharmonicity of the potential and analyzed how quantum dynamic depends on the wave packet impact parameter and beam's angular divergence. We found that the extrema of the electron trajectory period, considered as a function of the impact parameter, determine the shape of the caustic pattern. All wave packet probability densities have multiple maxima generated by a self-interference. Their sum, that represents probability density of an ensemble, was found to depends strongly on the beam angular divergence. For small divergence, most peaks of different wave packets are aligned causing wavelike behavior of the ensemble. For moderate angular divergence maxima of some wave packets, are aligned with minima of others, resulting in the emergence of the classical caustic pattern. We have shown and experimentally confirmed that the only indication that the observed caustic pattern is generated by the quantum dynamics is a slight systematic shift of the corresponding caustic maxima.

We investigate a reduced scaling full-potential DFT method based on the multiple scattering theory (MST) code MuST, which is released online (https://github.com/mstsuite/MuST) very recently. First, we test the accuracy by calculating structural properties of typical body-centered cubic (BCC) metals (V, Nb, and Mo). It is shown that the calculated lattice parameters, bulk moduli, and elastic constants agree with those obtained from the VASP, WIEN2k, EMTO, and Elk codes. Second, we test the locally self-consistent multiple scattering (LSMS) mode, which achieves reduced scaling by neglecting the multiple scattering processes beyond a cut-off radius. In the case of Nb, the accuracy of 0.5 mRy/atom can be achieved with a cut-off radius of 20 Bohr, even when small deformations are imposed on the lattice. Despite that the calculation of valence states based on MST exhibits linear scaling, the whole computational procedure has an overall scaling of about , due to the fact that the updating of Coulomb potential scales almost as . Nevertheless, it can be still expected that MuST would provide a reliable and accessible way to large-scale first-principles simulations of metals and alloys.

Calculations are presented demonstrating controlled intermolecular communication between dopant molecules via lattice phonons in a cryogenic matrix. By simulating a time-resolved coherent anti-Stokes Raman scattering (tr-CARS) experiment on an I2-molecule ‘receiver,’ its reception from a Ca-atom ‘transmitter’ of coherent lattice waves in a compressed linear chain of Ar atoms is monitored. A short pulse resonant with an atomic transition in Ca arrives at the sample before the tr-CARS pulses, and the coherent response of the host medium to the electronic excitation sends a rarefaction wave through the lattice. The arrival of this propagating distortion at the I2 molecule is detected through the tr-CARS difference signal, the difference between tr-CARS signals with and without the Ca pulse. A normal-mode analysis of the equilibrium configuration is used to construct an initial Gaussian wave packet, which is propagated by locally quadratic approximations to the relevant many-body nuclear Hamiltonians. Results are presented for both resonant and sub-resonant excitation of the Ca atom. Calculations show a difference signal that turns on abruptly as the lattice wave generated by resonant Ca excitation reaches the I2 chromophore. † Special number of Molecular Physics in honor of Robert A. Harris.