Quantum mechanics wave packet dynamics

As emphasized in this review, time-resolved X-ray diffraction of dynamical nonequilibrium structures involve, at any given time, the signal from a distribution of structures, as described by quantum mechanical wave-packets in the case of pure states. Thus, the interpretation of experimental data is nontrivial. For isolated molecules, direct inversion of diffraction data is possible in some cases as illustrated in this review. This approach might be useful also for reactions in the liquid phase - for the short-time dynamics before interaction with the solvent plays an important role. 

Fig. 2. Average rotational energy transfer (ARET) for different collision energies. Solid line quantum mechanical wave packet propagation using the MCTDH method (from Ref. [22]) dashed line MQCB method (equation (47)) dotted line classical dynamics.

Figure 4 shows two typical trajectories which trace the quantum mechanical wave-packet motion. They are started on the symmetric stretch line with different initial momenta pointing into the exit channels. The two trajectories represent dissociation into H + OD and D + OH, respectively. What was found for the quantum dynamics is more clearly demonstrated by the classical trjectories the H + OD dissociation is faster and the oscillations of both trajectories around the minimum energy path clearly shows the vibrational excitation of the fragments. Indeed, if we compute the time-evolution of the bondlength expectation values th d) from the bifurcated packets it is found that they resemble closely the classical trajectories (as can be expected from Ehrenfest s theorem). The wave-packet motion shows that the dissociation proceeds as can be anticipated classically. 

Molecular spectroscopy offers a fiindamental approach to intramolecular processes [18, 94]. The spectral analysis in temis of detailed quantum mechanical models in principle provides the complete infomiation about the wave-packet dynamics on a level of detail not easily accessible by time-resolved teclmiques. 

To obtain a first impression of the nonadiabatic wave-packet dynamics of the three-mode two-state model. Fig. 34 shows the quantum-mechanical probability density P (cp, f) = ( (f) / ) (p)(cp ( / (f)) of the system, plotted as a function of time t and the isomerization coordinate cp. To clearly show the... 

An example where insight into the detailed mechanism has been achieved is seen in the work by Woeste s group (Daniel et al., 2003). They combined femtosecond pump-probe experiments, ab initio quantum calculations and wave-packet dynamics simulations in order to decipher the reaction dynamics that underlie the optimal laser fields for producing the parent molecular ion and minimizing fragmentation when CpMn(CO)3 is photoionized (Cp = cyclopentadienyl) ... 

For systems with just one or two degrees of freedom, the explicit consideration of time-dependent wave packets as defined in Eqs. (22) and (24) can be illuminating. The book by Schinke on photodissociation dynamics, for example, contains nice examples.If the problem involves three or more nuclear degrees of freedom, on the other hand, a reduced description, which condenses the information carried by the wave packet, is desirable. Such reduced descriptions are obtained by integrating the probability density over part of the degrees of freedom. The relevance of reduced descriptions for complex systems is based on the fact that an experimental measurement will not yield the complete quantum mechanical wave function, but rather partially integrated information, e.g. the population of an electronic state or... 

Chemisorption of N2 [286, 632-640] and H2 [634] on Fe in relation to NH3 synthesis has been the subject of quantum-mechanical calculations. The dynamics of N2 chemisorption has been simulated using a semiclassical wave packet technique [641]. The simulations agree with the molecular beam experiments in the conclusion that vibrational excitation is of some importance. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

The first volume contained nine state-of-the-art chapters on fundamental aspects, on formalism, and on a variety of applications. The various discussions employ both stationary and time-dependent frameworks, with Hermitian and non-Hermitian Hamiltonian constructions. A variety of formal and computational results address themes from quantum and statistical mechanics to the detailed analysis of time evolution of material or photon wave packets, from the difficult problem of combining advanced many-electron methods with properties of field-free and field-induced resonances to the dynamics of molecular processes and coherence effects in strong electromagnetic fields and strong laser pulses, from portrayals of novel phase space approaches of quantum reactive scattering to aspects of recent developments related to quantum information processing. 

In the standard theory of quantum mechanics, two kinds of evolution processes are introduced, which are qualitatively different from each other. One is the spontaneous process, which is a reactive (unitary) dynamical process and is described by the Heisenberg or Schrodinger equation in an equivalent manner. The other is the measurement process, which is irreversible and described by the von Neumann projection postulate [26], which is the rigorous mathematical form of the reduction of the wave packet principle. The former process is deterministic and is uniquely described, while the latter process is essentially probabilistic and implies the statistical nature of quantum mechanics. 

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