Quantum trajectory

The results of the previous section have already established that classical chaos and quantum mechanics are not incompatible in the macroscopic limit. The question then naturally arises whether observed quantum mechanical systems can be chaotic far from the classical limit This question is particularly significant as closed quantum mechanical systems are not chaotic, at least in the conventional sense of dynamical systems theory (R. Kosloff et.al., 1981 1989). In the case of observed systems it has recently been shown, by defining and computing a maximal Lyapunov exponent applicable to quantum trajectories, that the answer is in the affirmative (S. Habib et.al., 1998). Thus, realistic quantum dynamical systems are chaotic in the conventional sense and there is no fundamental conflict between quantum mechanics and the existence of dynamical chaos. 

An alternative way to obtain the quantum trajectories is by formulating the Bohmian mechanics as a Newtonian-like theory. Then, Equation 8.29 gives rise to a generalized Newton s second law ... 

These new trajectories are the so-called reduced quantum trajectories [30], which are only explicitly related to the system reduced density matrix. The dynamics described by Equation 8.42 leads to the correct intensity (time evolution of which is described by Equation 8.40) when the statistics of a large number of particles are considered. Moreover, Equation 8.42 reduces to the well-known expression for the velocity held in Bohmian mechanics, when there is no interaction with the environment. 

It has been demonstrated that the quantum trajectory method is well suited to describe the motion along the y-direction as long as the real solution is not substantially bifurcating. 

Recently [8-11] an alternative treatment to mix quantum mechanics with classical mechanics, based on Bohmian quantum trajectories was proposed. Briefly, the quantum subsystem is described by a time-dependent Schrodinger equation that depends parametrically on classical variables. This is similar to other approaches discussed above. The difference comes from the way the classical trajectories are calculated. In our approach, which was called mixed quantum-classical Bohmian (MQCB) trajectories, the wave packet is used to define de Broglie-Bohm quantum trajectories [12] which in turn are used to calculate the force acting on the classical variables. 

Thus, within the Bohmian formulation of quantum mechanics, quantum trajectories move according to the usual Hamilton s equations, subject to the additional quantum potential defined in equation (3). An ensemble of quantum particles at positions (x(t),X(t)) distributed initially according to... 

In the fourth part, we study the effect of Cl on IC. It was applied to study the TtTt ->nTT transition of the pyrazine molecule. In this nonadiabatic process, the Cl of the TCK and nir PESs is believed to play a major role in the nonadiabatic fs transition. In fact, the Cl has been widely proposed to play the key factor in an IC, and quantum trajectory calculations have been used to calculate the IC rates [45]. However, this method cannot properly take into account of the initial conditions of the population and coherence of the system created by the fs pumping laser. In this chapter, we propose to develop a method to calculate the IC with conical intersections. It should be known that for the IC between Si and So in most molecules (in these cases, the energy gap between and So is of several eV), the surface crossings do not take place due to the anharmonic effect in the two PESs. Thus, the Cl should not play any role in these cases. We have proposed one method to calculate the IC rate of mt of the pyrazine molecule. The... 

The first two books in the series are (a) Aromaticity and Metal Clusters and (b) Quantum Trajectories. A two-book set on Concepts and Methods in Modern Theoretical Chemistry, edited by Swapan Kumar Ghosh and Pratim Kumar Chattaraj, is the new addition to this series. The first book focuses on the electronic structure and reactivity of many-electron systems and the second book deals with the statistical mechanical treatment of collections of such systems. 

Pratim Kumar Chattaraj earned a BSc (Honors) and an MSc from Burdwan University and a PhD from the Indian Institute of Technology (IIT), Bombay, India, and then joined the faculty of the IIT, Kharagpur, India. He is now a professor with the Department of Chemistry and also the convener of the Center for Theoretical Studies there. In the meantime, he visited the University of North Carolina, Chapel HiU, as a postdoctoral research associate and several other universities throughout the world as a visiting professor. Apart from -------------- teaching. Professor Chattaraj is involved in research on density functional theory, the theory of chemical reactivity, aromaticity in metal clusters, ab initio calculations, quantum trajectories, and nonlinear dynamics. He has... 

Three novel approaches to the simulation of NA dynamics of large chemical systems have been presented [20-22]. The approaches extend the standard quantum-classical NA MD to incorporate quantum effects of the solvent subsystem that have been traditionally treated by classical mechanics. These effects include quantum trajectory branching (wave packet splitting), loss of quantum coherence directly related to the Franck-Condon overlap contribution to the NA transition probability, and ZPE of nuclear motion that contributes to the NA coupling and must be preserved during the equilibration of the energy released by the NA transition. 

In the classical path formulation of collision problems one divides the phase space in two sets a classical x and a quantum y The dynamical motion in the classical phase space is as mentioned in the introduction governed by an effective or average potential. However, it has been known for many years that it is necessary also to introduce a "symmetrization procedure" of the S-matrix elements or transition probabilities obtained by solving the equations (1, 2). This symmetrization procedure which has been motivated by results obtained using first order perturbation technique (for further details see ref. [1]) has recently been rationalized by the quantum trajectory derivation by Muckerman et al. C2] ... 

P. Holland. Quantum Trajectories. Ed. P Chattarai (Taylor Francis/CRC, Boca Raton, 2010) Chap. 5. 

In the variational multi-configuration Gaussian wavepacket (vMCG) method [10, 11, 59-63] the basis functions follow coupled quantum trajectories whereby the mean positions and momenta are treated as variational basis-function parameters that evolve according to the Dirac-Frenkel principle applied to the TDSE. Each basis function directly simulates quantum phenomena in a rigorous way, and the method thus promises much faster convergence than classical-trajectory-based methods due to a better sampling of the phase space. 

In contrast with grid-based methods, there is no need to first build analytical models for the potential energy surface, which makes Gaussian-based methods easier to use. In addition, an approximate description based on a limited number of quantum trajectories can provide a more intuitive interpretation of the reaction mechanism than the evolution of a wavepacket on a grid. However, increasing the size and complexity of the system may require an even more approximate description such as semi-classical trajectories presented below. 

For other hydrogen-transfer reactions, satisfactory ab initio potential energy surfaces [115, 116] and approximate quantum trajectory calculations of isotope effects [136] are also available in a few instances, but for the most part experimental measurements have been compared with values calculated on the basis of empirical or semi-empirical potential surfaces, making use of transition-state theory, and discussion has centred on the pros and cons of the surface used and the type of tunnelling correction made [3, 19]. 

Since the stabihty parameters A. depend on the quantum trajectory X( r), the effective classical trajectory of instanton is the solution of SS ff/SX = 0, namely. 

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