Schrodinger equation time-dependent wave function

Abstract. Cross sections for electron transfer in collisions of atomic hydrogen with fully stripped carbon ions are studied for impact energies from 0.1 to 500 keV/u. A semi-classical close-coupling approach is used within the impact parameter approximation. To solve the time-dependent Schrodinger equation the electronic wave function is expanded on a two-center atomic state basis set. The projectile states are modified by translational factors to take into account the relative motion of the two centers. For the processes C6++H(1.s) —> C5+ (nlm) + H+, we present shell-selective electron transfer cross sections, based on computations performed with an expansion spanning all states ofC5+( =l-6) shells and the H(ls) state. 

We start from the time-dependent Schrodinger equation for the wave-function describing the evolution of the unperturbed molecular system,... 

Presently, we assume that we have a time-dependent wave function, 10(/)>, and that it is normalized to unity. Furthermore, we require that 10(/)> reduces to the time-independent wave function, O), in the limit of no perturbation. The time-independent wave function, O), is the solution to the time-independent Schrodinger equation and 0) is normalized. Therefore, for an exact state we write the time-dependent wave function as [50,51]... 

In Chapter 3 we investigated the development in time of a decaying state, expressed in terms of the time-independent eigenfunctions satisfying a system of two coupled differential equations, resulting from the separation of the Schrodinger equation in parabolic coordinates. In this analysis we obtained general expressions for the time-dependent wave function and the probability amplitude. 

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... 

A fundamental theoretical issue for computational photochemistry is the treatment of the hop (nonadiabatic) event. One needs to add the time propagation of the solutions of the time-dependent Schrodinger equation for electronic motion to the classical propagation of the nuclei, thus obtaining the populations of each adiabatic state. The time-dependent wave function for electronic motion is just a time-dependent configuration interaction vector ... 

We can write the time-dependent wave function, T (t))/ solving the Schrodinger equation... 

The variational principle for the action integral is derived starting from the observation that, if the time-dependent wave function 4/(r, 0 is a solution of the time-dependent Schrodinger equation, then it corresponds to a stationary point of the quantum-mechanical action integral ... 

If the wave function at a certain time is given by a linear combination such as that in Example 16.19, the time-dependent wave function is determined by this function and the time-dependent Schrodinger equation. 

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... 

The time-dependent differential equation for the wave function W(x,t) bears the name of Schrodinger. For a one-dimensional case it is given by... 

Postulate III gives the time evolution equation for the wave function iff (time-dependent Schrodinger equation Ht / = ih ), using the energy operator Hamiltonian H). 

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. 

Wavefunctions describing time-dependent states are solutions to Schrodinger s time-dependent equation. The absolute square of such a wavefunction gives a particle distribution function that depends on time. The time evolution of this particle distribution function is the quantum-mechanical equivalent of the classical concept of a trajectory. It is often convenient to express the time-dependent wave packet as a linear combination of eigenfunctions of the time-independent hamiltonian multiplied by their time-dependent phase factors. 

Time-dependent density functional theory (TDDFT), in contrast, applies the same philosophy as ground-state DFT to time-dependent problems. Here, the complicated many-body time-dependent Schrodinger equation is replaced by a set of time-dependent single-particle equations whose orbitals yield the same time-dependent density. We can do this because the Runge-Gross theorem proves that, for a given initial wave function, particle statistics and interaction, a given time-dependent density can arise from, at most, one time-dependent external potential. This means that the time-dependent potential (and all other properties) is a functional of the time-dependent density. 

The vibrational structure may be explained as follows For each state of a molecule there is a wave function that depends on time, as well as on the internal space and spin coordinates of all electrons and all nuclei, assuming that the overall translational and rotational motions of the molecule have been separated from internal motion. A set of stationary states exists whose observable properties, such as energy, charge density, etc., do not change in time. These states may be described by the time-independent part of their wave functions alone. Their wave functions are the solutions of the time-independent Schrodinger equation and depend only on the internal coordinates q = q q-,. .. of all electrons and the internal coordinates Q = Qi, Qi,. . . of all nuclei. 

Jackson [15,16] proposes a quantum-mechanical theory for nonelastic scattering of particles as a result of interaction with surface phonons. Translational degrees of freedom of the gas particle are presented as a time-dependent wave packet. The wave functions describing the scattering of the particles satisfy a Schrodinger-type equation with a potential of interaction between flie gas particle and the solid surface, which depends both on time and temperature. In this model the Hamiltonian of the system is... 

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

Equation (4.a) states that the wave function must obey the time-dependent Schrodinger equation with initial condition /(t = 0) = < ),. Equation (4.b) states that the undetermined Lagrange multiplier, x t), must obey the time-dependent Schrodinger equation with the boundary condition that x(T) = ( /(T))<1> at the end of the pulse, that is at f = T. As this boundary condition is given at the end of the pulse, we must integrate the Schrodinger equation backward in time to find X(f). The final of the three equations, Eq. (4.c), is really an equation for the time-dependent electric field, e(f). 

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