Atoms time-dependent wave functions

To obtain a rough estimate of the excitation transfer cross section, consider first the adiabatic approximation, neglecting the Coriolis coupling (the so-called rotating-atom approximation ). The transfer is then associated with mere interference between gerade and ungerade states. As an example, the 2 time-dependent wave function can be written as... 

The time evolution of molecular systems - i.e. systems containing particles of atomic dimensions ( electrons, nuclei etc) - can be adequately described within the framework of time-dependent quantum mechanics. All information about the system is contained in the time dependent wave function 0(r,r2,...,rN t) wherein the Ti are the position vectors (possibly containing also a spin component) for the i-th particle and t is the time. The wave function is a solution of the time-dependent Schrodinger equation... 

To illustrate the exchange of the phase information between the atomic transition and the multipole field, consider the electric dipole Jaynes-Cummings model (34). Assume that the field consists of two circularly polarized components in a coherent state each. The atom is supposed to be initially in the ground state. Then, the time-dependent wave function of the system has the form [53]... 

It describes the radiated power, while the atomic electron undergoes the dynamics. Another important physical quantity of interest is the ATI spectrum, which corresponds to the energy density spectrum of the electron in the continuum. It can be calculated by projecting the simulated, time-dependent wave function Itf (O) that evolves from the initially unperturbed ground state of the atom I 0) = < (r — — oo)) under the influence of the external laser pulse onto free-electron continuum states transition amplitudes involved are obtained via... 

In quantum mechanics the probability Viit) of finding an atomic system at the time t in the quantum state 11) is described by time-dependent wave functions. If one wants to find out whether a system is with certainty [Pi(r) = 1] in a well-defined quantum state 1> one has to perform a measurement that, however, changes this state of the system. Many controversial opinions have been published on whether it is possible to perform experiments with a single atom in such a way that its initial state and a possible transition to a well-defined final state can be unambiguously determined. 

In quantum mechanics the probability (t) of finding an atomic system at the time t in the quantum state 1> is described by time-dependent wave functions. If one wants to find out whether a system is with certainty = 1] in... 

The Time-Dependent Wave Functions of the Hydrogen Atom... 

The position and energy of each electron surrounding the nucleus of an atom are described by a wave function, which represents a solution to the Schrodinger wave equation. These wave functions express the spatial distribution of electron density about the nucleus, and are thus related to the probability of finding the electron at a particular point at an instant of time. The wave function for each electron, F(r,6,), may be written as the product of four separate functions, three of which depend on the polar coordinates of the electron... 

Banerjee and Harbola [69] have worked out a variation perturbation method within the hydrodynamic approach to the time-dependent density functional theory (TDDFT) in order to evaluate the linear and nonlinear responses of alkali metal clusters. They employed the spherical jellium background model to determine the static and degenerate four-wave mixing (DFWM) y and showed that y evolves almost linearly with the number of atoms in the cluster. 

Quantum mechanics is based on the wave nature of all atomic particles. In a H atom, an electron orbits around the nucleus (a proton) electron energies, or energy states, can be conveniently described in terms of a wave function, P(x , y, z, t), which depends on particle space coordinates, x, y, z, and time t. Stable states having well-defined (discrete) energies can be represented as the product of a sinusoidal time-dependent term of angular frequency co, and a time-independent wave function l/(x, y, z) ... 

Abstract We review and further develop the excited state structural analysis (ESSA) which was proposed many years ago [Luzanov AV (1980) Russ Chem Rev 49 1033] for semiempirical models of r r -transitions and which was extended quite recently to the time-dependent density functional theory. Herein we discuss ESSA with some new features (generalized bond orders, similarity measures etc.) and provide additional applications of the ESSA to various topics of spectrochemistry and photochemistry. The illustrations focus primarily on the visualization of electronic transitions by portraying the excitation localization on atoms and molecular fragments and by detaiUng excited state structure using specialized charge transfer numbers. An extension of ESSA to general-type wave functions is briefly considered. 

Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body Schrodinger equation, its basic variable is the one-body electron density, n(r,t). The advantages are clear The many-body wave-function, a function in a 3A-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector r. The standard way to obtain n r,t) is with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated. 

The first two chapters serve as an introduction to quantum theory. It is assumed that the student has already been exposed to elementary quantum mechanics and to the historical events that led to its development in an undergraduate physical chemistry course or in a course on atomic physics. Accordingly, the historical development of quantum theory is not covered. To serve as a rationale for the postulates of quantum theory, Chapter 1 discusses wave motion and wave packets and then relates particle motion to wave motion. In Chapter 2 the time-dependent and time-independent Schrodinger equations are introduced along with a discussion of wave functions for particles in a potential field. Some instructors may wish to omit the first or both of these chapters or to present abbreviated versions. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

A serious problem in LA-ICP-MS described in the literature on many occasions is the time-dependent elemental fraction (so-called ablation fractionation ) occurring during laser ablation and the transport process of ablated material, or during atomization and ionization processes in the inductively coupled plasma.20-22 Numerous papers focus on the study of fraction effects in LA-ICP-MS as a function of experimental parameters applied during laser ablation (such as laser energy, laser power density, laser pulse duration, wave length of laser beam, ablation spot size,... 

A relaxation process will occur when a compound state of the system with large amplitude of a sparse subsystem component evolves so that the continuum component grows with time. We then say that the dynamic component of this state s wave function decays with time. Familiar examples of such relaxation processes are the a decay of nuclei, the radiative decay of atoms, atomic and molecular autoionization processes, and molecular predissociation. In all these cases a compound state of the physical system decays into a true continuum or into a quasicontinuum, the choice of the description of the dissipative subsystem depending solely on what boundary conditions are applied at large distances from the atom or molecule. The general theory of quantum mechanics leads to the conclusion that there is a set of features common to all compound states of a wide class of systems. For example, the shapes of many resonances are nearly the same, and the rates of decay of many different kinds of metastable states are of the same functional form. 

For the case of a central field, the energy of an atom does not depend on magnetic quantum number m/. This means that the energy level, characterized by quantum numbers n and /, is degenerated 2/+1 times. For a pure Coulomb field there exists additional (hydrogenic) degeneration the energy of such an atom does not depend on /. Wave function (1.14) may... 

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