Wave function time-dependent

In tills weakly coupled regime, ET in an encounter complex can be described approximately using a two-level system model [23]. As such, tlie time-dependent wave function is... 

They unfold a connection between parts of time-dependent wave functions that arises from the structure of the defining equation (2) and some simple properties of the Hamiltonian. 

The mixed-state character of a trajectory outside a non-adiabatic region is a serious weakness of the method. As the time-dependent wave function does not... 

The simplified theory allows the time-dependent wave function to be calculated rapidly for any specified laser field. However, controlling the dynamics of the charge carriers requires the answer to an inverse question [18-22]. That is, given a specific target or objective, what is the laser field that best drives the system to that objective Several methods have been developed to address this question. This section sketches one method, valid in the weak response (perturbative) regime in which most experiments on semiconductors are performed. 

The time-dependent wave function, t /(r, R, t e(t)), can be expanded in terms of the field-dependent adiabatic electronic eigenfunctions of Eq. (25) ... 

Writing E2 = E + AE, the general time-dependent wave function of the two-level system may be formulated as... 

One may write the time-dependent wave function in the polar form, viz.,... 

But in fact, Y (or the more general time-dependent wave function Y, which is the product of two functions, one involving the time alone and the other the coordinate alone) was difficult to interpret physically, because the idea of the... 

The states correspond to wave packet controlled in the far past and in the far future, respectively. Let us see what this means. In the absence of external time-dependent fields, the scattering component of the time-dependent wave function i/r(f) can be expanded in terms of either of the two sets of scattering states for example, those with incoming boundary conditions... 

Heather, R.W. and Metiu, H. (1986). A numerical study of the multiple Gaussian representation of time dependent wave functions of a Morse oscillator, J. Chem. Phys. 84, 3250-3259. 

We evaluate the time-dependent expectation values using the time-dependent wave-function, O >,... 

While F[] = [ ], the mean values of the time derivative are not equal. Even if H is independent of time, these values are constants J2, n> > and E, respectively. These constants can be reconciled by inserting a gauge potential y = E — JA nif-i into A, determined by equating the time-dependent phase factors. This also preserves the normalization (T T) = (4> 4>) = 1 for the time-dependent wave functions, so that the definition of correlation energy = ( H T — ) remains un-... 

In Bohmian formalism it may be argued that the reaction system, considered closed, is described at all times by an equation H Sf = IA > in the time-dependent wave function T(/l, />, 6, I)). The product states 4,a- 4,b and as well as the valence state are special solutions of this equa-... 

Assuming that the field is in near resonance or on resonance with transitions from the initial bound state [i ) to the continuum (see Fig. 10.1a), we expand the lull time-dependent wave function as ... 

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]... 

The time-dependent wave function ip(x,y,z t) for the internal motion is expanded as a superposition of solutions of the form (2.15) in the following way... 

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. 

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... 

We consider, for simplicity, a spin /2 nucleus, with eigenfunctions a and /3, as described in Section 2.3. Note that a and /3 represent stationary eigenstates, but this does not imply that a nuclear spin must reside only in one of them. A general, time-dependent wave function can be constructed as a linear combination (or... 

We determine the contributions to the linear, quadratic and cubic response equations by expanding the time-dependent wave function, 0 ), and the time-dependent operator, to the appropriate perturbation order. Finally, we collect the terms related to a given order of the perturbation and we have that the terms relevant for... 

By virtue of the 1-1 correspondence established above (for a given Pq), the time-dependent density determines the external potential uniquely up to within an additive purely time-dependent function. The potential, on the other hand, determines the time-dependent wave function, which can therefore be considered as a functional of the time-dependent density, unique up to within a purely time-dependent phase a(t) ... 

Since there is a 1-1 mapping between time-dependent wave functions, P(r), and time-dependent densities, n(r, t), the corresponding density functional... 

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