Phase factors evolution

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. 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

The next level in the evolution of understanding of the electromagnetic phase is to consider that all optical phases are derived from the non-Abelian Stokes theorem (482), so all optical phases originate in the phase factor... 

The master equation evolves the classical degrees of freedom on single adiabatic surfaces with instantaneous hops between them. Each single (fictitious) trajectory represents an ensemble of trajectories corresponding to different environment initial conditions. This choice of different environment coordinates for a given initial subsystem coordinate will result in different trajectories on the mean surface the average over this collection of classical evolution segments results in decoherence. Consequently, this master equation in full phase space provides a description in terms of fictitious trajectories, each of which accounts for decoherence. When the approximations that lead to the master equation are valid, this provides a useful simulation tool since no oscillatory phase factors appear in the trajectory evolution. 

We remark that the simulation scheme for master equation dynamics has a number of attractive features when compared to quantum-classical Liouville dynamics. The solution of the master equation consists of two numerically simple parts. The first is the computation of the memory function which involves adiabatic evolution along mean surfaces. Once the transition rates are known as a function of the subsystem coordinates, the sequential short-time propagation algorithm may be used to evolve the observable or density. Since the dynamics is restricted to single adiabatic surfaces, no phase factors... 

The time-dependent KS Eq. 3 can be used to simulate the full time evolution of the density. However, in this review article we shall exclusively consider the linear response TDDF(R)T approach. Its starting point is Eq. 3 with the time-independent potential vSCT(ri). In this case, the orbitals phase factor that oscillates in time... 

Figure 1. The creation, evolution, and detection of wave packets. The pump laser pulse pump (black) creates a coherent superposition of molecular eigenstates at t — 0 from the ground state I k,). The set of excited-state eigenstates N) in the superposition (wave packet) have different energy-phase factors, leading to nonstationary behavior (wave packet evolution). At time t = At the wave packet is projected by a probe pulse i probe (gray) onto a set of final states I kf) that act as a template for the dynamics. The time-dependent probability of being in a given final state f) is modulated by the interferences between all degenerate coherent two-photon transition amplitudes leading to that final state.

We can also have /-coupling evolution from 1+ to I+Sz or from I SZ to I-, but the coherence order (1 and -1, respectively) does not change. Because ZQC and DQC do not undergo/-coupling evolution, I+S+ will stay as I+S+ andl+S- will stay as I+S- during a delay (times a phase factor for DQ or ZQ chemical-shift evolution) and the coherence order (5 and 3, respectively) will not change. 

The data are processed in a different way, combining the echo and antiecho FIDs for each t value to regenerate the real and imaginary (cosine modulated and sine modulated) FIDs. Then the data is processed just like States data. How this is done can be seen if we examine the phase factors that result from evolution during t ... 

In the absence of the perturbation, the time-evolution operator produces only a dynamical phase factor, which cancels out for expectation values. In the presence of the perturbation, it is assumed that an expansion in powers of the perturbation exists such that... 

Except for a phase factor, the time evolution is given by an oscillating position shift, X. Using this and (2.176) in (2.187) yields the result... 

Simpler and more manageable expressions are obtained in the limit where the dynamics of the bath is much faster than that of the system. In this limit the functions A/ (Z) (i.e. the bath coiTelations functions C(f) of Eq. (10.121) or Cm (Z) of Eq. (10.124)) decay to zero as t oo much faster than any characteristic system timescale. One may be tempted to apply this limit by substituting r in the elements Omn(t — 1 ) in Eq. (10.141) by zero and take these terms out of the integral over r, however this would be wrong because, in addition to their relatively slow physical time evolution, non-diagonal elements of d contain a fast phase factor. Consider for example the integral in first term on the right-hand side of (10.141),... 

After the time zero, the wave functions i> and 1 2 have the classical free evolution (in the absence of interaction) characterized by the phase factors exponential of - i(E/)iOt. But we must add to the phase factors a real factor, exponential of - (r/2)t, representing the spontaneous decay to the final state with the decay rate F, inverse of the lifetime (the decay rate F/2 on the amplitudes gives the decay rate r on the intensities, when you square the amplitudes). So we obtain the wavefunction of the excited state to time t positive ... 

It has to be emphasized that only an ultrashort laserpulse can create a localized wave packet as displayed in the figure. The longer the pulse, the more the prepared state will be delocalized in coordinate space and thus resemble a single stationary scattering state of the molecule. The time evolution of such a state is given by a phase factor and thus the whole idea of pump/probe spectroscopy is lost. 

Wavefunctions describing time-dependent states are solutions to Schrodinger s time-dependent equation. The absolute square of such a wavefunction gives a particle distribution function that depends on time. The time evolution of this particle distribution function is the quantum-mechanical equivalent of the classical concept of a trajectory. It is often convenient to express the time-dependent wave packet as a linear combination of eigenfunctions of the time-independent hamiltonian multiplied by their time-dependent phase factors. 

The time evolution of the electronic eigenfunctions cpi obtained at time t = 0 from a quantum chemical code is given by their phase factor Q- dR(. )) xhe corresponding eigenenergies Ei(R(t)) are functions of the nuclear coordinates due to the propagating nuclear wavefunctions. To keep track of the actual phase factor of the electronic wavefunction from time step to time step this phase factor needs to be calculated recursively by utilizing the phase of the previous time step ... 

When the time-dependent perturbation H (f) is absent, and the parameters p have their equilibrium value po (i.e. when do = 0), (12.3.4) ensures that do remain zero and the variation function % be time-independent (remembering that the trivial phase factor exp(-iEotlh) has been discarded). When the perturbation is applied, starting we suppose at f = -oo, the initial conditions are do = 0 and VH = 0, and the time evolution is determined to first order by solving (12.3.5) for d. We imagine the perturbation to be turned on infinitely slowly, corresponding to an adiabatic process, and seek the best variational approximation to W at time t. On putting do = 0 in (12.3.5), we have... 

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