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22 A superoperator is an operator that operates on ordinary operators. 231t does not remain diagonal if the atoms are injected in a quantum superposition of upper and lower levels. 24 For simplicity, we assume that the temperature is low enough for the average number of thermal photons to be negligible.

(mmmr) = (mmrm)



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It is very difficult to control experimentally the time between successive atoms in an atomic beam. Here we follow Filipowicz et aL [209] and assume that the time interval tp has an exponential probability distribution (ljlp) exp( -tjlp), with mean lp. Assuming that the fluctuations of each tp are statistically independent. exp( tpL) and p(ti + tint) in (8.117) will not be correlated and the average yields the product of two separate averages: one over the present t p, the other over previous tp's on which p(ti + tint) alone depends (see the appendix of [209] for a formal proof of this). Calling p such an averaged p, we can write the evolution of the field in the micromaser as

(mr) = 2 (mm) (rrrm) = (mmrr) = 2 (rmmr)





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May 13, 2018 · In order to get rid of the evaluation-mark from output file,. ... See how easily you can convert a ...Duration: 2:02 Posted: May 13, 2018

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(8.118) The steady state of the mapping (8.118) is given by25 (8.119) Using that the ({nl, In))-matrix element of (8.116) is

Only mr (rrrm) = (rrmr)

(nILpln)

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amount of steric bulk required at the active site in order to observe high enantioselectivity between the two monomer faces. The monomer and various transition metal ligands contribute to the steric bulk at the active site. For propene polymerization, the enantioselectivity in the rst monomer unit added is low for R methyl, appreciable for R ethyl, and very high for R isobutyl or phenyl. Enantioselectivity of the rst addition is enhanced for larger-sized monomers such as styrene and vinylcyclohexane compared to propene and for larger-sized halogen ligands. The various steric effects act in a cooperative manner. Thus, addition of the rst monomer unit for propene polymerization is as isoselective as all subsequent monomer units when TiI3 is used instead of TiCl3 . The Cossee Arlman mechanism as originally proposed has a weakness the back- ip is required to explain isoselective placement since the two active (coordination) sites are assumed to be enantiotopic. However, the structure of the traditional Ziegler Natta heterogeneous initiators is not suf cently understood to either support or reject the assumption of enantiotopic sites. Further, even if the sites are enantiotopic, there is no overwhelming reason why the polymer chain is more stable at one site than the other which is the rationale for the back- ip. The mechanism of isoselectivity with various metallocene initiators is much better understood since these are initiators whose molecular structures are well-established [Busico and Cipullo, 2001; Busico et al., 1997, 1999; Cavallo et al., 1998; Ewen, 1999; Rappe et al., 2000; Resconi et al., 2000]. Considerable advancements in understanding heterogeneous Ziegler Natta initiators occur if one assumes that the active sites in these initiators mimic those in metallocene initiators. Two types of metallocene initiators offer possible models

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for the isoselective active sites in heterogeneous Ziegler Natta initiators, the C1 - and C2 symmetric metallocenes. These will be discussed in greater detail in Secs. 8-5a-2 and 8-5a-4. The following gives a brief overview with speci c application to the heterogeneous Ziegler Natta initiators: 1. C1 -symmetric initiators have a pair of diastereotopic (nonequivalent) sites; one site is sterically crowded and enantioselective, and the other site is less crowded and nonselective. The propagating polymer chain always prefers the less crowded site, but monomer coordination and migratory insertion occur at the more crowded enantioselective site. The polymer chain then back- ips to the less crowded site. This model offers a rationale for the back- ip of the polymer chain the polymer chain is less stable at the more crowded site. 2. C2 -symetric initiators have a pair of equivalent homotopic sites, both of which prefer the same monomer enantioface, that is, both sites prefer the re enantioface or both prefer the si face. Isoselective propagation proceeds with or without migratory insertion since coordination and insertion of monomer at either site give the same stereochemical result.

we find that (Problem 8.8)

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