Quantum evolution

Figure A3.13.14. Illustration of the quantum evolution (pomts) and Pauli master equation evolution (lines) in quantum level structures with two levels (and 59 states each, left-hand side) and tln-ee levels (and 39 states each, right-hand side) corresponding to a model of the energy shell IVR (liorizontal transition in figure...

The results of the quantum simulations for cases A, B and C are shown in the lower two panels in Fig. 3. The corresponding classical phase portraits shown reinforce our inferences from the stability diagram no stabilization for A while larger islands exist for C as compared with B. However, the ionized fraction as calculated from the quantum evolution supports the contrary result that there is more stabilization for A as compared with B. Case C is the most stable which is at least consistent with the classical prediction. What is the origin of this discrepancy ... 

A remarkable fact is that, in spite of all fluctuations and fractal properties exhibited by quantum motion, strong empirical evidence has been obtained that the quantum evolution is very stable, in sharp contrast to the extreme sensitivity to initial conditions that is the very essence of classical chaos [2]. 

A complete and detailed analysis of the formal properties of the QCL approach [5] has revealed that while this scheme is internally consistent, inconsistencies arise in the formulation of a quantum-classical statistical mechanics within such a framework. In particular, the fact that time translation invariance and the Kubo identity are only valid to O(h) have implications for the calculation of quantum-classical correlation functions. Such an analysis has not yet been conducted for the ILDM approach. In this chapter we adopt an alternative prescription [6,7]. This alternative approach supposes that we start with the full quantum statistical mechanical structure of time correlation functions, average values, or, in general, the time dependent density, and develop independent approximations to both the quantum evolution, and to the equilibrium density. Such an approach has proven particularly useful in many applications [8,9]. As was pointed out in the earlier publications [6,7], the consistency between the quantum equilibrium structure and the approximate... 

The elements of information summarized above are sufficient for our needs. We move on to examine the measurement problem from a perspective advanced in Refs. [4] and [5] and further developed here. Because the quantum evolution of probing and probe is entangled in Hilbert space and the events in real-space elicit results in measurements, this junction zone is a Fence. 

Consider a system that can exist in two spectroscopic states V and [Eq.(19)]. Quantum evolution of this system under the following nonlinear Hami1tonian,... 

In order to eliminate an unimportant global phase in our two-state approximation, it is convenient to represent the quantum evolution of the electronic wave function by Bloch variable. The Bloch variables x, y and z axe defined via the density matrix pmn — A,tiA according to... 

We show how the quantum-classical evolution equations of motion can be obtained as an approximation to the full quantum evolution and point out some of the difficulties that arise because of the lack of a Lie algebraic structure. The computation of transport properties is discussed from two different perspectives. Transport coefficient formulas may be derived by starting from an approximate quantum-classical description of the system. Alternatively, the exact quantum transport coefficients may be taken as the starting point of the computation with quantum-classical approximations made only to the dynamics while retaining the full quantum equilibrium structure. The utility of quantum-classical Liouville methods is illustrated by considering the computation of the rate constants of quantum chemical reactions in the condensed phase. 

The Wigner form of the quantum evolution operator iLw X ) in (62) for the equation of motion for W X, X2,t) can be rewritten in a form that is convenient for the passage to the quantum-classical limit. Recalling that the system may be partitioned into S and S subspaces, the Poisson bracket operator A can be written as the sum of Poisson bracket operators acting in each of these subspaces as A Xi) = A xi) + A Xi). Thus, we may write... 

M. Eden, A. Brinkmann, H. Luthman, L. Eriksson, and M. H. Levitt, Determination of molecular geometry by high-order multiple-quantum evolution in solid-state NMR, J. Magn. Reson., 144 (2000) 266-279. 

Environment or surroundings interaction is not, however, the only way of introducing decay or decoherence into quantum evolution. Measurement processes in which the system state couples to that of the measurement device are often described in terms of instantaneous projective measurements, which also de-... 

Millot and Man have provided detailed tables of solutions of Eqs. (57) and (58) for all cases (i-iv) and (a-d), with respect to dwell times Au and Au (1 + k). Here, we describe two solutions, the first giving intuitive results in which axes correspond directly to the (split-U) experimental parameters and the second resulting in a unified representation of all the MQMAS experiments. The first choice corresponds to option (a) and takes k = (1 + /c). In addition, A i(l + K) is the increment time since it corresponds to both single- and multiple-quantum evolution and Eq. (54) is used for the offset values. This results in the following relations for spins 3/2 and 5/2 ... 

When analytical solutions are not known and the approximate analytical methods give results of limited applicability, the numerical methods may be a solution. Let us first discuss a method based on the diagonalization of the second-harmonic Hamiltonian [48,49]. As we have already said, the two parts of the Hamiltonian Ho and Hi given by (55), commute with each other, so they are both constants of motion. The //0 determines the total energy stored in both modes, which is conserved by the interaction ///. This means that we can factor the quantum evolution operator... 

Figure 4. Quantum evolution of the Q function for the fundamental (outer contour plots) and the second-harmonic mode (inner plots) at six time moments for initial coherent states with 0(1 = 6,0(2 - 5.0 0. Solution obtained by quantum numerical method.

Figure 6. Classical trajectory simulation of quantum evolution of the Q function for the same initial conditions and interaction times as in Fig. 4. In our simulation 10,000 trajectories were calculated.

The generation of double-quantum coherence requires an antiphase disposition of coupling vectors, which here develop during a period A = l/2Jcc- This is provided in the form of a homonuclear spin-echo to make the excitation independent of chemical shifts. The ti period represents a genuine double-quantum evolution period in which these coherences evolve at the sums of the rotating-ffame frequencies of the two coupled spins, that is, at the sums of their offsets... 

Further details of multiple-quantum evolution can be found in section 5.3 of Ernst, Bodenhausen and Wokaun Principles of NMR in One and Two Dimensions (Oxford University Press, 1987). 

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