Wave-packet delocalization

The first two terms arise from modes that are extended over the protein. In proteins the size of cytochrome c, myoglobin, and GFP, which range, respectively, from 103 to 228 amino acids, almost all extended modes are fracton modes. The number of phonon modes in proteins of this size is very small, only 24 modes from 4 to 13 cm 1 for GFP, the largest of the three proteins. We cannot isolate the dynamics of a wave packet in terms of just these modes for an object this small. Thus, the contribution of extended modes to thermal conduction in proteins arises almost entirely from delocalized fractons. We give an expression for this contribution below. 

Figure 8. The square of the nuclear amplitude on the lower adiabatic Jahn-Teller surface at several propagation times in the A = 2.1 system. The motion proceeds in a counterclockwise direction. The wave packet does not become delocalized until after it has completed one pseudorotation in the excited state at t, 2(2ir/< )).

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. 

Since both experiments (picosecond and femtosecond) were carried out with low temperatures in the molecular beam of approximately 50 K, the initial vibrational state is assumed to be a coherent linear superposition of the three lowest delocalized vibrational eigenfunctions Xv=o-2 of the X state, which are very close in energy [62, 75, 79, 81, 380, 387]. Therefore, the Boltzmann distribution yields nearly identical populations of these three eigenfunctions at the relevant temperature. The linear superposition leads to an initial wave packet that is localized in one of the equivalent minima of the X state ... 

When a molecule is excited by an ultrashort laser pulse with an appropriate center frequency, a localized wave packet can be created in the excited electronic state because of the excitation of a coherent superposition of many vibrational-rotational states. It follows from fundamental laws that the d3mamics of molecular wave packets is governed by a time-dependent Schrodinger equation (eqn 2.29), where H is the relevant Hamiltonian of the given molecule. Because molecular potential-energy surfaces are anharmonic, this molecular wave packet tends to spread both in position (coordinates) and in momentum. However, in addition to expansion or defocusing, the wave packet also suffers delocalization at a certain instant of time. Coherent quantum... 

The right-hand side of the above equation describes the error introduced by the Hartree approximation. The error vanishes if the Hamiltonian is separable, and it becomes small if the functions W i and W2 are almost constant over the width of the single-particle functions

time-independent Hartree approach. The time-dependent wave packet is more or less localized, whereas the eigenstates are usually very delocalized. 

Quantum localization behavior in K-G lattice has been studied by many researchers in terms of four atom lattice with periodic function, notably by Proville [106], delocalization and spreading behavior of wave-packets by Flach et al. [38], dimer case for targeted energy transfer by Aubry et al. [43], Here, we present a generalized method for any number of sites and quanta without periodic boundary condition to show the QB states. In K-G lattice, it is important to calculate the critical time of redistribution of quanta under various physical conditions. It is the time when the temporal evolution of the number of quanta first meets or tends to meet. 

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