Quantum time evolution

Peskin U, Miller W H and Ediund A 1995 Quantum time evolution in time-dependent fields and time-independent reactive-scattering calculations via an efficient Fourier grid preconditioner J. Chem. Phys. 103 10 030... 

T. J. Park and J.C. Light Unitary quantum time evolution by iterative Lanczos reduction. J. Chem. Phys. 85 (1986) 5870-5876... 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

Broeckhove, J., Feyen, B., Lathouwers, L., Arickx, F., and Van Leuven, P. (1990). Quantum time evolution of vibrational states in curve-crossing problems, Chem. Phys. Lett. 174, 504-510. 

Note that the invariance of quantum observables under unitary transformations has enabled us to represent quantum time evolutions either as an evolution of the wavefunction with the operator fixed, or as an evolution of the operator with constant wavefunctions. Equation (2.1) describes the time evolution of wavefunctions in the Schrodinger picture. In the Heisenberg picture the wavefunctions do not evolve in time. Instead we have a time evolution equation for the Heisenberg operators ... 

Table 10.1 The Schrodinger, Heisenberg, and interaction representations of the quantum time evolution.

The brackets indicate a proper thermal average, C is a constant, is the real part of z, and f = —1. The derivative takes into account that the lineshape is usually displayed in derivative mode. The correlation function SxSx t)) is evaluated by the quantum time evolution of the electron spin under the influence of the reorientation of the spin probe according to the equation of motion [31] ... 

The calculation of the spectral density function which describes the quantum equilibrium structure is a difficult problem, but is easier than the computation of the full quantum time evolution of a large many-body system. 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

Figure A3.13.7. Continuation of the time evolution for the CH eln-omophore in CHF after 90 fs of irradiation (see also figure A3,13,6). Distanees between tire eontoiir lines are 10, 29, 16 and 9 x 10 rr in the order of the four images shown. The averaged energy of the wave paeket eorresponds to 9200 em (roughly 6300 em absorbed) with a quantum meehanieal imeertainty of +5700 enC (from [97]).

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

Let us consider the time evolution of a quantum system, which satisfies the time-dependent Schiodinger equation [55]... 

Quantum Cellular Automata (QCA) in order to address the possibly very fundamental role CA-like dynamics may play in the microphysical domain, some form of quantum dynamical generalization to the basic rule structure must be considered. One way to do this is to replace the usual time evolution of what may now be called classical site values ct, by unitary transitions between fe-component complex probability- amplitude states, ct > - defined in sncli a way as to permit superposition of states. As is standard in quantum mechanics, the absolute square of these amplitudes is then interpreted to give the probability of observing the corresponding classical value. Two indepcuidently defined models - both of which exhibit much of the typically quantum behavior observed in real systems are discussed in chapter 8.2,... 

A more elegant model of quantum mechanical computers was introduced by Feynman ([feyii85], [feynSb]). Although the formalism is similar to Benioff s - both approaches seek to define an appropriate quantum mechanical Hamiltonian H whose time evolution effectively represents the execution of a desired computation... 

The form of the action principle given above was first applied to quantum mechanics to describe the time evolution of pure states (i.e. 

Quantum mechanics is essential for studying enzymatic processes [1-3]. Depending on the specific problem of interest, there are different requirements on the level of theory used and the scale of treatment involved. This ranges from the simplest cluster representation of the active site, modeled by the most accurate quantum chemical methods, to a hybrid description of the biomacromolecular catalyst by quantum mechanics and molecular mechanics (QM/MM) [1], to the full treatment of the entire enzyme-solvent system by a fully quantum-mechanical force field [4-8], In addition, the time-evolution of the macromolecular system can be modeled purely by classical mechanics in molecular dynamicssimulations, whereas the explicit incorporation... 

A quantum algorithm can be seen as the controlled time evolution of a physical system obeying the laws of quantum mechanics. It is therefore of utmost importance that each qubit may be coherently manipulated, between arbitrary superpositions, via the application of external stimuli. Furthermore, all these manipulations must take place well before its quantum wave function, thus the information it encodes, is corrupted by the interaction with external perturbations. The need to properly isolate qubits but, at the same time, to rapidly... 

Here, t/(f) is the reduced time evolution operator of the driven damped quantum harmonic oscillator. Recall that representation II was used in preceding treatments, taking into account the indirect damping of the hydrogen bond. After rearrangements, the autocorrelation function (45) takes the form [8]... 

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