Effective operators time-independent operator

Consider a time-independent operator A whose matrix elements, yf a, /3 d (both expectation values and transition moments), in the space fl we wish to compute. This goal is to be achieved by transforming the calculation from 0 into one in O, resulting in an effective operator a whose matrix elements, taken between appropriate model eigenfunctions of an effective Hamiltonian h, are the desired As we now discuss, numerous possible definitions of a arise depending on the type of mapping operators that are used to produce h and on the choice of model eigenfunctions. 

Although rules and procedures are a necessary and indeed essential aspect of safety, they need to be regularly reviewed and updated in the light of feedback from operational experience. Unfortunately, such feedback loops become less and less effective with time, and hence need to be reviewed regularly, preferably by an independent third party... 

A systematic study of the salting-out precipitation process was made to obtain characteristic particle size and productivity data within wide range of the initial supersaturations. The appropriate supersaturations were created by changing two independent parameters the initial solution concentration and the solubility. The operational time was fixed at a given system but in limited cases the effect of the operational time on the particle size was also investigated. 

Because we are working to linear order in iAL, we have used the fact that uUo,s) can contribute only higher powers of iL = ILq + iAL. Fligher powers of linear approximation. Effectively, it can be seen that the operator Ur(0, s) = e ° + 0(/AL), where O(iAL) indicates terms of order /AL and where we have used Eq. [99] and the time independence of /Lq. We also note that by definition, iLo/eq = 0. Thus only the first term in the Taylor series expansion of 17 (0, s) survives, yielding = /eq and we obtain Eq. [103]. 

This demonstrates again the special form of the propagator, a doubly periodic operator times an unitary operator defined by a time independent effective Hamiltonian. 

This is the Markovian memory-less approximation to the Master Equation. In this approximation, the effective time evolution operator becomes independent of t and the integral may be extended to infinity. It is also consistent to assume that the system lost memory of the initial state of the reservoir, whatever this was. In the limit when Uq is calculated in perturbation theory and pq(0) = 0, we obtain the conventional Born-Markov time evolution which has a long and successful history. 

The use of van Vleck s contact transformation method for the study of time-dependent interactions in solid-state NMR by Floquet theory has been proposed. Floquet theory has been used for studying the spin dynamics of MAS NMR experiments. The contact transformation method is an operator method in time-independent perturbation theory and has been used to obtain effective Hamiltonians in molecular spectroscopy. This has been combined with Floquet theory to study the dynamics of a dipolar coupled spin (I = 1/2) system. 

Effective Hamiltonians and Effective Operator Definitions Corresponding to a Time-Independent Operator A... 

As we have seen previously, closed shell systems with an even number of electrons are invariant under the inversion of the time-coordinate. In the time-independent theory this could be related to the anti-unitary operator, K, that commutes with the DCB Hamiltonian. Its effect on the Tj operator is given by... 

On the other hand, approximations to Eq. (8) and time-integration techniques, suitable especially for time-independent Hamiltonians, under the requirement of only a few degrees of freedom and short-time evolution, have been developed and applied extensively in connection with grid-type techniques (see Section 2), by focusing on appropriate algebraic expansions of fhe exponenfial form. For example, such a approach is effected by the split-operator method [4] and references there in. 

Solution of the Lippmann-Schwinger-like equation in Brillouin-Wigner form, equation (55), for the reaction operator followed by solution of the eigenvalue problem (49) for the effective hamiltonian given in equation (52) is entirely equivalent to the solution of the time-independent Schrodinger equation, equation (1), for the state a. Furthermore, although recursion leads to the expansion (56), equation (55) remains valid when the series expansion does not converge. Equation (55) can be written... 

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