Evolution time

The invariance of transition probabilities under the action of a symmetry operator T, 

Operators that induce transformations in space satisfy eq. (2) and are therefore unitary operators with the property / T = 1. An operator that satisfies eq. (3) is said to be antiunitary. In contrast to spatial symmetry operators, the time-reversal operator is anti-unitary. Let U denote a unitary operator and let T denote an antiunitary operator. 

unitary operators are linear operators, but an antiunitary operator is antilinear. 

Time evolution in quantum mechanics is described, in the Schrodinger representation, by the Schrodinger time-dependent equation 

---

The time evolution of the wavefiinction is described by the differential equation... 

A differential equation for the time evolution of the density operator may be derived by taking the time derivative of equation (Al.6.49) and using the TDSE to replace the time derivative of the wavefiinction with the Hamiltonian operating on the wavefiinction. The result is called the Liouville equation, that is. 

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... 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The methods described here are all designed to detennine the time evolution of wavepackets that have been previously defined. This is only one of several steps for using wavepackets to solve scattering problems. The overall procedure involves the following steps ... 

Vibrational motion is thus an important primary step in a general reaction mechanism and detailed investigation of this motion is of utmost relevance for our understanding of the dynamics of chemical reactions. In classical mechanics, vibrational motion is described by the time evolution and l t) of general internal position and momentum coordinates. These time dependent fiinctions are solutions of the classical equations of motion, e.g. Newton s equations for given initial conditions and I Iq) = Pq. 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

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... 

Figure A3.13.6. Time evolution of the probability density of the CH cliromophore in CHF after 50 fs of irradiation with an excitation wave number = 2832.42 at an intensity 7q = 30 TW cm. The contour...

The later time evolution is shown in Figrne A3.13.7 between 90 and 100 fs, and m Figrne A3.13.8, between 390 and 400 fs, after the beginning of the excitation (time step t j)- Tln-ee observations are readily made first,... 

Figure A3.13.7. Continuation of the time evolution for the CH eln-omophore in CHF after 90 fs of irradiation (see also figure A3,13,6). Distanees between tire eontoiir lines are 10, 29, 16 and 9 x 10 rr in the order of the four images shown. The averaged energy of the wave paeket eorresponds to 9200 em (roughly 6300 em absorbed) with a quantum meehanieal imeertainty of +5700 enC (from [97]).

In view of the foregoing discussion, one might ask what is a typical time evolution of the wave packet for the isolated molecule, what are typical tune scales and, if initial conditions are such that an entire energy shell participates, does the wave packet resulting from the coherent dynamics look like a microcanonical... 

Figure A3.13.il. Illustration of the time evolution of redueed two-dimensional probability densities I I and I I for the exeitation of CHD between 50 and 70 fs (see [154] for further details). The full eurve is a eut of tire potential energy surfaee at the momentary absorbed energy eorresponding to 3000 em during the entire time interval shown here (as6000 em, if zero point energy is ineluded). The dashed eurves show the energy uneertainty of the time-dependent wave paeket, approximately 500 em Left-hand side exeitation along the v-axis (see figure A3.13.5). The vertieal axis in the two-dimensional eontour line representations is...

The time evolution of is shown in figure A3.13.12 for the field free motion of wave packets for CHD2T and CHDT2 prepared by a preceding excitation along the> -axis. 

Figure A3.13.12. Evolution of the probability for a right-handed ehiral stmetnre (fiill eiirve, see ( equation (A3,13.69))) of the CH eliromophore in CHD2T (a) and CHDT2 ( ) after preparation of ehiral stnietures with multiphoton laser exeitation, as diseussed in the text (see also [154]). For eomparison, the time evolution of aeeording to a one-dimensional model ineluding only the bending mode (dashed enrve) is also shown. The left-hand side insert shows the time evolution of within the one-dimensional ealeulations for a longer time interval the right-hand insert shows the time evolution within the tln-ee-dimensional ealeulation for the same time interval (see text).

Figure A3.13.13. Illustration of the time evolution of redueed two-dimensional probability densities I and I I for the isolated CHD2T (left-hand side) and CHDT, (right-hand side) after 800 fs of free evolution. At time 0 fs the wave paekets eorresponded to a loeafized, ehiral moleeular stnieture (from [154]). See also text and figure A3.13.il.

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

B1.3.2.4 TIME EVOLUTION OF THE THIRD ORDER POLARIZATION BY WAVE MIXING ENERGY LEVEL (WMEL) DIAGRAMS. THE RAMAN SPECTROSCOPIES CLASSIFIED... 

The main cost of this enlianced time resolution compared to fluorescence upconversion, however, is the aforementioned problem of time ordering of the photons that arrive from the pump and probe pulses. Wlien the probe pulse either precedes or trails the arrival of the pump pulse by a time interval that is significantly longer than the pulse duration, the action of the probe and pump pulses on the populations resident in the various resonant states is nnambiguous. When the pump and probe pulses temporally overlap in tlie sample, however, all possible time orderings of field-molecule interactions contribute to the response and complicate the interpretation. Double-sided Feymuan diagrams, which provide a pictorial view of the density matrix s time evolution under the action of the laser pulses, can be used to detenuine the various contributions to the sample response [125]. 

---