Phase-space wavepackets

Appendix C Four-Point Correlation Function Expression for Fluorescence Spectra Appendix D Phase-Space Doorway-Window Wavepackets for Fluorescence Appendix E Doorway-Window Phase-Space Wavepackets for Pump-Probe Signals References... 

Phase-space wavepackets for nuclear motions have been applied to the interpretation of nonlinear optical measurements using the Liouville space... 

APPENDIX E DOORWAY-WINDOW PHASE-SPACE WAVEPACKETS FOR PUMP-PROBE SIGNALS... 

In the full quantum mechanical picture, the evolving wavepackets are delocalized functions, representing the probability of finding the nuclei at a particular point in space. This representation is unsuitable for direct dynamics as it is necessary to know the potential surface over a region of space at each point in time. Fortunately, there are approximate formulations based on trajectories in phase space, which will be discussed below. These local representations, so-called as only a portion of the FES is examined at each point in time, have a classical flavor. The delocalized and nonlocal nature of the full solution of the Schtddinger equation should, however, be kept in mind. 

Finally, Gaussian wavepacket methods are described in which the nuclear wavepacket is described by one or more Gaussian functions. Again the equations of motion to be solved have the fomi of classical trajectories in phase space. Now, however, each trajectory has a quantum character due to its spread in coordinate space. 

To return to the simple picture of vertical excitation, the question remains as to how a wavepacket can be simulated using classical trajectories A classical ensemble can be specified by its distribution in phase space, Pd(p,Q), which gives the probability of finding the system of particles with momentum p and position q. In conUast, it is strictly impossible to assign simultaneously a position and momentum to a quantum particle. 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

III. Wigner Wavepackets in Phase Space The Doorway—Window Picture... 

Now the overlap of these wavepackets is calculated in phase space. Note that this is a fully quantum mechanical picture, and no classical approximations were made. Our only assumption is that the excitation and gating processes are well separated. This form, however, allows the development... 

Similar to the correlation measurements discussed in the previous sections, we can define the doorway and window wavepackets and write the signal as their overlap in phase space. The derivation is presented in Appendix E, and we have... 

APPENDIX D PHASE-SPACE DOORWAY-WINDOW WAVEPACKETS FOR FLUORESCENCE... 

Similar to the fluoiesence discussed in Appendix D, we can define doorway and window wavepackets and write the signal as their phase-space overlap. Assuming that the delay time between the probe and the pump pulses is to, we obtain Eqs. (4.5)-(4.8). 

The optimization of the pump pulse leads to a localization of the phase space density around the intermediate target. The intermediate target operator can be represented in the Wigner representation [Eq. (20)] by a minimum uncertainty wavepacket ... 

In other words, the FMS, method in principle, can converge to an exact description of the nuclear dynamics if an infinite number of nuclear basis functions is included that span the entire phase space of an infinite number of electronic states. The approximation consists of allowing only relatively few of these basis functions to have a nonvanishing coefficient Cf it). The algorithm attempts to select the nuclear basis functions that are the most important for the description of the wavepackets. 

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