Wave packet localized

The matter wave function is formed as a coherent superposition of states or a state ensemble, a wave packet. As the phase relationships change the wave packet moves, and spreads, not necessarily in only one direction the localized launch configuration disperses or propagates with the wave packet. The initially localized wave packets evolve like single-molecule trajectories. 

Anharmonic effects lead to a destruction of the initially localized wave-packet with the consequence that the transient signal is damped. However, there is the possibility for the wavepacket to regain its initial shape after long times resulting in so-called revivals [see, for example, Alber and Zoller (1991)]. The revival period Trev contains additional information about the shape of the potential. 

Finally, we like to mention that equivalent to the conventional energy frame KHD formulation, the time-dependent theory of Raman scattering is free from any approximations except the usual second order perturbation method used to derive the KHD expression. When applied to resonance and near resonance Raman scattering, the time-dependent formulation has shown advantages over the static KHD formulation. Apparently, the time-dependent formulation lends itselfs to an interpretation where localized wave packets follow classical-like paths. As an example of the numerical calculation of continuum resonance Raman spectra we show in Fig. 6.1-7 the simulation of the A, = 4 transitions (third overtone) of D excited with Aq = 488.0 nm. Both, the KHD (Eqs. 6.1-2 and 6.1-18) as well as the time-dependent approach (Eqs. 6.1-2 and 6.1-19) very nicely simulate the experimental spectrum which consists mainly of Q- and S-branch transitions (Ganz and Kiefer, 1993b). 

The initial wavefunction is constructed as the direct product of a localized wave packet for R, R), as defined in Eq. (11), and a specific state, ) of the system... 

The concept of quantized energy can be pursued further to describe the motion of energy in free space as a localized wave packet. Therefore it is at the origin of quantum electrodynamics. 

Kosloff, 1994) have also been used to find the complex energies for compound-state resonances. A localized wave packet [i.e., a coherent superposition state, Eq. (4.7)], P(O) is initially placed in the bound region of the potential energy surface and propagated in time to give ( l (0) (r)), which is C t) in Eq. (4.16). If I (O) is a superposition of resonant states, it can be considered a zero-order state (see chapter 4) and can be written as... 

Since the survival probability may be ditficult to measure, some decay analyses discuss other quantities, such as the nonescape probability from a region of space [57, 58], the probability density at chosen points of space [25, 59, 60], the flux [61-63], and the arrival time [64]. For initially localized wave packets, there is no major discrepancy between survival probability and the nonescape probability [3, 57, 59, 65-67]. Examination of densities, fluxes, or arrival time distributions may be interesting since a new variable is introduced (we shall see later some applications), but at the price of losing the simplicity and directness of the survival probability. 

In effect, the short laser pulse launches, or projects, a localized wave packet onto the excited-state PES, as depicted conceptually in Figure 15.Bl. 

The article is organized as follows in Sec. 2 we outline the scheme of a pump/probe experiment which is performed with two ultrashort laser pulses. Then the excitation mechanism which prepares the system in a coherent superposition of eigenstates in the transition-state region of the respective potential surface is described (Sec. 3). This localized wave packet evolves along the reaction path, as described in Sec. 4. In Sec. 5 it is shown how the wave-packet motion can be probed. A short summary concludes this chapter. 

As can be taken from the figure a localized wave packet is prepared in the short pulse excitation process. Its location is close to the potential barrier which separates the two product channels H -f- OD and D -f- Oif, i.e. the packet is found in the vicinity of the transition-state re on. Nevertheless parts of the initial wave packet have already moved towards the exit channels. Since the if-atom is lighter than the D-atom it... 

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. 

Ultrashort pulses are able to prepare localized wave packets which (for short times) move in the average classically. This motion clearly defines a reaction path for the case that the reaction starts right at the transition state and the molecular fragments evolve into the arrangement channels. In this sense the laser induced process is a half collision , since the first part of a full collision where the atomic and molecular species approach each other is missing. 

The two spin states share the occupation equally. In the weak coupling limit is slightly less than unity. The number of holes in the conduction band is equal to rtf, also with one-half of the amount in each spin state. Since a hole with spin up leaves an unpaired electron with spin down and vice versa, the ground state may be looked upon as a singlet state formed by the f electron and a local wave packet formed by band electrons. 

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... 

Using ultrafast pump excitation we can launch such a localized wave-packet in the transition state region of a chemical reaction and then probe the temporal evolution toward the products (Bernstein and Zewail, 1988 Zewail, 1996, 2000). 

Figure 8.3 A schematic two-dimensional view of the potential energy surface and wave-packet dynamics in the ultrafast photodissociation of Hgl2 [adapted from Voth and Hochstrasser (1996), Zewail (1996)]. The transition state for the I + Hgl reaction is along the bisector, dashed line, with the lowest barrier at the bottom of the potential along that line. The UV excitation creates a localized wave-packet along the bisector. The center of the packet is displaced from the saddle point to a compressed configuration along the symmetric stretch. During the dissociation the wave-packet bifurcates, as shown, and each component is followed in the figure. It shows the coherent vibrational motion in the Hg—I well. ...

F. Spreading of wave-packets. In an anharmonic potential, a localized wave-packet will, in general, spread. Show this for the simplest case, a constant potential. Take an initial state that is a Gaussian like Eq. (8.1), but allow its width a to depend on time. By substituting this function in the Schrddinger time-dependent equation show that it is a solution. Thereby show how a increases with time. See C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, New York, John Wiley, 1977, complement GI. 

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