Wavefunction time-dependent

Fig. 2. Schematic configuration space for the reaction AB + CD — A + BCD. R is the radial coordinate between the center-of-mass of the two diatoms, and r is the vibrational coordinate of the reactive AB diatom. I denotes the interaction region and II denotes the asymptotic region. The shaded regions are the absorption zones for the time-dependent wavefunction to avoid boundary reflections. The reactive flux is evaluated at the r = rB surface.

Similar to Eq. (2), the time-dependent wavefunction is expanded in terms of the BF parity-adapted rotational basis functions ... 

An ansatz is then made for the time-dependent wavefunction l o (t)) depending on time-dependent coefficients which are expanded in the orders of the perturbations Ef or The wavefunction coefficients in each order of the perturbation have to be determined from the time-dependent Schrddinger equation or an equivalent time-dependent variation principle. Expressions... 

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

The formulation of the extended Wigner-Weisskopf scheme proceeds along lines similar to those employed in the derivation of the decay law for a single level8 in Section III. The time-dependent wavefunction is displayed as a superposition of the eigenstates of HR + (see eq. (10-8)) ... 

Projection of the time-dependent wavefunction on the scattering states... 

The channel-resolved photoelectron distribution of the time-dependent wavefunction ip (t) at the end of the external pulses is obtained by taking the square module of the projection of fit) with the scattering functions V e ... 

We now briefly describe the way in which the nonadiabatic event (surface hop) can be described in on the fly dynamics methods. We can represent the time-dependent wavefunction in the Cl space as a vector ... 

In Section 2.1 we derived the expression for the transition rate kfi (2.22) by expanding the time-dependent wavefunction P(t) in terms of orthogonal and complete stationary wavefunctions Fa [see Equation (2.9)]. For bound-free transitions we proceed in the same way with the exception that the expansion functions for the nuclear part of the total wavefunction are continuum rather than bound-state wavefunctions. The definition and construction of the continuum basis belongs to the field of scattering theory (Wu and Ohmura 1962 Taylor 1972). In the following we present a short summary specialized to the linear triatomic molecule. 

As in Section 2.5 we expand the total time-dependent wavefunction as... 

The introduction of the time-transformed operator and the time-dependent wavefunction leads to... 

The time-dependent wavefunction is expanded in the parity-adapted rotational basis functions as. 

The result of a molecular dynamics simulation is a time dependent wavefunction (quantum dynamics) or a swarm of trajectories in a phase space (classical dynamics). To analyze what are the processes taking the place during photodissoeiation one can directly look at these. This analysis is, however, impractical for systems with a high dimensionality. We can calculate either (juantities in the time domain or in the energy domain, fn the time domain survival probabilities can be measured by pump-probe experiments [44], in the energy domain the distribution of the hydrogen kinetic energy can be experimentally obtained [8]. 

We can also find the time-dependent wavefunction explicitly. Combining Eqs (2.107), (2.105), and (2.94), and converting again the sum over k to an integral leads to... 

What is the quantum mechanical analog of this approach Consider the simple example that describes the decay of a single level coupled to a continuum. Fig. 9.1 and Eq. (9.2). The time-dependent wavefunction for this model is (Z) = Ci (t) 11)+ Ci(t ) l, where the time-dependent coefficients satisfy (cf. Eqs (9.6) and (9.7))... 

Consider a system characterized by agiven Hamiltonian operator., an orthonormal basis (/) (also denoted n ) that spans the corresponding Hilbert space and a time dependent wavefunction h (i)—a normalized solution of the Schrodinger equation. The latter may be represented in terms of the basis ftmctions as... 

In order to calculate the transition probability we have to start with the time-dependent wavefunction which is given in accordance with Eq. (6.79)... 

The time-dependent wavefunction which describes the electronic... 

Expanding the time-dependent wavefunction for the interacting system in this basis of unperturbed functions, Eqs. (2) and (3), and using the machinery of time-dependent perturbation theory yields the differential cross section cr q)dq as function of the momentum transfer q... 

Figure 17. Effective geometry of the model. The x axis corresponds to the quasi-one-dimensional time-dependent wavefunction.

The potentials which will be considered here are stepwise constant. In each region of these potentials the time-dependent wavefunction is a linear combination of solutions of the wave equation for a particle interacting with an electromagnetic field of vector potential with A(t) [10] ... 

Time-dependent perturbation theory proceeds by expanding a general time-dependent wavefunction... 

P9.18 To address this time-dependent problem, we need a time-dependent wavefunction, made up from... 

In the case of a time-dependent wavefunction, the corresponding relationship becomes E—>ihd/dt, and Schrodinger s wave equation is given by... 

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as terms that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. When this condition is satisfied, the representation of the time-dependent wavefunction as a superposition of Hamiltonian eigenfunctions can be used to determine the time dependence of the expansion coefficients. If equation (A 1.1.3 9) is substituted into the time-dependent Schrodinger equation... 

Close inspection of equation (A 1.1.45) reveals that, under very special circumstances, the expectation value does not change with time for any system properties that correspond to fixed (static) operator representations. Specifically, if the spatial part of the time-dependent wavefunction is the exact eigenfunction /. of the Hamiltonian, then c. (0) = 1 (the zero of time can be chosen arbitrarily) and all other c (0) = 0. fhe second term clearly vanishes in these cases, which are known as stationary states. As the name implies, all observable properties of these states do not vary with time. In a stationary state, the energy of the system has a precise value (the corresponding eigenvalue of //) as do observables that are associated with operators that commute with //. For all other properties (such as the position and momentum). 

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