Phase factor time-dependent

What is the effect of a time delay on the density matrix Each off-diagonal element gets multiplied by a phase factor that depends on the length of time of the delay and the energy difference between the two energy levels that are connected by that transition ... 

So long as the field is on, these populations continue to change however, once the external field is turned off, these populations remain constant (discounting relaxation processes, which will be introduced below). Yet the amplitudes in the states i and i / do continue to change with time, due to the accumulation of time-dependent phase factors during the field-free evolution. We can obtain a convenient separation of the time-dependent and the time-mdependent quantities by defining a density matrix, p. For the case of the wavefiinction ), p is given as the outer product of v i) with itself. 

An alternative perspective is as follows. A 5-frmction pulse in time has an infinitely broad frequency range. Thus, the pulse promotes transitions to all the excited-state vibrational eigenstates having good overlap (Franck-Condon factors) with the initial vibrational state. The pulse, by virtue of its coherence, in fact prepares a coherent superposition of all these excited-state vibrational eigenstates. From the earlier sections, we know that each of these eigenstates evolves with a different time-dependent phase factor, leading to coherent spatial translation of the wavepacket. 

The time-dependent structure factor S k,t), which is proportional to the intensity I k,t) measured in an elastic scattering experiment, is a measure of the strength of the spatial correlations in the ordering system with wavenumber k at time t. It exliibits a peak whose position is inversely proportional to the average domain size. As the system phase separates (orders) the peak moves towards increasingly smaller wavenumbers (see figure A3.3.3. 

Figure A3.3.3 Time-dependent structure factor as measured tlnough light scattering experiments from a phase...

With the above definitions, there is no additional overall phase factor to be included in (27). Eqs. (24)-(27) are the CSP approximation.Like TDSCF, CSP is a separable approximation, using a time-dependent mean potential for each degree of freedom. However, the effective potentials in CSP... 

In these early reactions the reactivities of the individual phases are important in determining the overall reaction rate. However, as the cement particles become more densely coated with reaction products, diffusion of water and ions in solution becomes increasingly impeded. The reactions then become diffusion-controUed at some time depending on various factors such as temperature and water—cement ratio. After about 1 or 2 days, ie, at ca 40% of complete reaction, the remaining unhydrated cement phases react more nearly uniformly. 

If the Hamilton operator, H, is independent of time, the time dependence of the wave function can be separated out as a simple phase factor. 

Proof. When the time-dependent Schrodinger equation is solved under adiabatic conditions, the upper, positive energy component has the coefficient the dynamic phase factor x C, where... 

The above spectral densities can be modified for the occurence of chain flexibility, and for the director being oriented at dLD w.r.t. the external BQ field in the L frame. For CD bonds located in the flexible chain, the effect of DF is reduced due to an additional averaging of the time dependent factor (/f g) by conformational transitions in the chain. Consequently, the spectral densities given in Eqs. (60)-(62) are modified by replacing Soc%0(Pm,q) by the segmental order parameter YCD of the C-D bond at a particular carbon site on the chain.146,147 As observed experimentally,148,149 the spectral densities in a flexible chain show a SqD dependence when DF dominate the relaxation rates. The general expression of Jm(co 0LD) due to DF in uniaxial nematic phases is given by... 

Adsorption kinetics involve a time-dependent process that describes the rate of adsorption of chemical contaminants on the solid phase. The standard chemical meaning of kinetics usually covers the study of the rate of reactions and molecular processes when transport is not a limiting factor however, this definition is not... 

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

In such a case, the time dependence of the state is carried in the phase factor exp(-itE/h) the spatial dependence appears in ]/(qj). 

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