Time-evolution operator function

Here, t/(f) is the reduced time evolution operator of the driven damped quantum harmonic oscillator. Recall that representation II was used in preceding treatments, taking into account the indirect damping of the hydrogen bond. After rearrangements, the autocorrelation function (45) takes the form [8]... 

In Ref. [9] we demonstrated how one approaches the DCL for the CC absorption cross section, Eq. (31). In a first step, the overall time evolution operator exp(iHcct/k) has to be replaced by the 5-operator 5i(t, 0) which includes the difference Hamiltonian of the excited CC state and of the ground-state. Then, the vibrational Hamiltonian matrix appearing in the exponent of 5i(t, 0) is replace by an ordinary matrix the time-dependence of which follows from classical nuclear dynamics in the CC ground-state. The time-dependence of the dipole moment d follows from intra chromophore nuclear rearrangement and changes of the overall spatial orientation. At last, this translation procedure replaces the CC state matrix elements of the 5-operator by complex time-dependent functions... 

The time evolution operator exp(—/HAf/ft) acting on ( ) propagates the wave function forward in time. A number of propagation methods have been developed and we will briefly describe the following the split operator method [91,94,95], the Lanzcos method [96] and the polynomial methods such as Chebychev [93,97], Newtonian [98], Faber [99] and Hermite [100,101]. A classical comparison between the three first mentioned methods was done by Leforestier et al. [102]. 

In developing the quantum amplitude for an optical process, it is necessary to determine the matrix elements of the time evolution operator, and to this end it is frequently expedient to invoke an expansion in terms of operators rather than the embedded time integrals of Eq. (28). The method of resolvent operators, which affords a framework for both perturbative and nonperturbative analysis [28-30], proceeds through the introduction of a retarded Green function, K+(t) = t/(f,O)0(f), together with its advanced counterpart, K (t) U(t,0)0( t),... 

The ealeulation of the time evolution operator in multidimensional systems is a formidable task and some results will be discussed in this section. An alternative approach is the calculation of semi-classical dynamics as demonstrated, among others, by Heller [M, and M], Marcus [M, 90], Taylor [91, 92], Metiu [93, 94] and coworkers (see also [M] as well as the review by Miller [95] for more general aspects of semiclassical dynamics). This method basically consists of replacing the 5-function distribution in the true classical calculation by a Gaussian distribution in coordinate space. It allows for a simulation of the vibrational... 

The analogy of the time-evolution operator in quantum mechanics on the one hand, and the transfer matrix and the Markov matrix in statistical mechanics on the other, allows the two fields to share numerous techniques. Specifically, a transfer matrix G of a statistical mechanical lattice system in d dimensions often can be interpreted as the evolution operator in discrete, imaginary time t of a quantum mechanical analog in d — 1 dimensions. That is, G exp(—tJf), where is the Hamiltonian of a system in d — 1 dimensions, the quantum mechanical analog of the statistical mechanical system. From this point of view, the computation of the partition function and of the ground-state energy are essentially the same problems finding... 

The time-evolution operator exp(— ) in the position representation is the Green function... 

Note that the wave operator used here has a very similar meaning as the time evolution operator U(0, -l-oo) in the time-dependent perturbation theory. Having the wave operator, we can then define an effective operator Hgff = Ho-f Veff which, if applied onto the model functions... 

Using the time-evolution operator (17.11), by means of the unitary transformation (18.11), we can determine the wave function in a moment t2 after the collision in terms of the wave function in the moment t. before the collision. 

S)Tmnetry of the Hamiltonian (p. 63) S)Tnmetry P (p. 72) time-evolution operator (p. 85) time-independent perturhation (p. 95) translational sjanmetry (p. 68) two-state model (p. 91) wave function evolution (p. 85) wave function matching (p. 82)... 

All wave functions addressed above are stationary , i.e. they do not change in time. In the quantum theoretical picture this means that they are eigenfunctions of the time evolution operator, which is the Hamiltonian of the system. The Hamiltonian is, in fact, the operator that describes the system in full, at all times. The eigenvalues of the Hamiltonian operating on the wave functions provide the energy of that state... 

The autocorrelation function is a very useful quantity, because it reflects the dynamics of the wave packets, that is how fast they depart from the region, where they were launched, how often they recur to the place of birth, for how long the wave packets remain compact and localized and on which timescale the bonds are broken, Eventually, when the entire wave packet has left the interaction region, i.e. when all molecules are dissociated, S t) becomes zero. Even though the initial wave packet is a real function, it becomes complex, because the time evolution operator is complex. Thus, S t) is a complex function and fulfills the symmetry relation... 

Time-Dependent Density Functional Theory 157 where the time-evolution operator, U, is defined by... 

Unfortunately, the expression (4.53) does not retain one of the most important properties of the Kohn-Sham time-evolution operator unitarity. In other words, if we apply (4.53) to a normalized wave-function the result will no longer be normalized. This leads to an inherently unstable propagation. 

The time-evolution operator for the Schrodinger wave function transforms the wave function from one time to another... 

Quantum dynamics can be treated equally easily with this path integral formalism. By making the replacement -> it/h, the Boltzmann operator exp(—yS//) turns into the time-evolution operator exp —itH/h). In other words, yS can be considered to be an imaginary time and the time-evolution operator can be written as the product of Q short-time propagators as in equation (3). With this, the Green s function is... 

Here i, sf,.., and iQ, if,... denote discretizations of the forward and backward path employed in the path integral representation of the forward and reverse time evolution operators, respectively. The influence functional has the structure... 

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