Operator time evolution

Its solution can be written in tenns of the time evolution operator t)... 

For themial unimolecular reactions with bimolecular collisional activation steps and for bimolecular reactions, more specifically one takes the limit of tire time evolution operator for - co and t —> + co to describe isolated binary collision events. The corresponding matrix representation of f)is called the scattering matrix or S-matrix with matrix elements... 

The calculation of the time evolution operator in multidimensional systems is a fomiidable 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 [86, 87 and 88], Marcus [89, 90], Taylor [91, 92], Metiu [93, 94] and coworkers (see also [83] as well as the review by Miller [95] for more general aspects of semiclassical dynamics). This method basically consists of replacing the 5-fimction distribution in the true classical calculation by a Gaussian distribution in coordinate space. It allows for a simulation of the vibrational... 

Equation (A3.13.54) legitimates the use of this semi-classical approximation of the molecule-field interaction in the low-pressure regime. Since /7j(t) is explicitly time dependent, the time evolution operator is more... 

The problem is then reduced to the representation of the time-evolution operator [104,105]. For example, the Lanczos algorithm could be used to generate the eigenvalues of H, which can be used to set up the representation of the exponentiated operator. Again, the methods are based on matrix-vector operations, but now much larger steps are possible. 

The time-evolution operator acts on each site state (Tj >,... 

To avoid discouraging the reader, we point out that only a few, basic concepts of path integrals are required. We review these here in a heuristic manner, beginning with the celebrated result of Feynman and Hibbs [40], which is that the time-evolution operator or Kernel, K = exp(—can be constructed using... 

The time-evolution operator U(t) for a single electron moving in a 3D-linear potential can be expressed in the form... 

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]... 

Sebastian et al. (SJG) concerned themselves with the time-evolution operator of the system, which satisfies the equation... 

Further evaluation of Eq. (2.35) requires an expression connecting 0(g)) (assumed to be nondegenerate) with Ox 5 ) (also assumed to be nondegenerate). This link is established via the interaction-picture time-evolution operator i.e. by an adiabatic switching oiHi ... 

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 time evolution operator propagates an arbitrary quantum state thanks to the non-zero matrix elements (Ve0ph)j m ,jm- The set ( )j(q)Qm(Q) has all electronic base states corresponding to all possible chemical species in the sense discussed above only because the generalized electronic diabatic set is complete. 

From (7.10) we can get the time evolution operator PV(t)P by use of a Fourier transform. These localized operators permit the construction of those portions of the time-evolved state vectors that lie within the subspace of P states. The influence of the remaining Q states occurs through the action of the operator M(co). 

Models for the dissipative dynamics can frequently be based on the assumption of fast decay of memory effects, due to the presence of many degrees of freedom in the s-region. This is the usual Markoff assumption of instantaneous dissipation. Two such models give the Lindblad form of dissipative rates, and rates from dissipative potentials. The Lindblad-type expression was originally derived using semigroup properties of time-evolution operators in dissipative systems. [45, 46] It has been rederived in a variety of ways and implemented in applications. [47, 48] It is given in our notation by... 

Here the time evolution operator including H4 was additionally introduced using the cyclic invariance of the trace with respect to the index 4. 

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