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So if the state is a mixture, the trace of p~ will be less than 1. Only when the state is pure is the trace of p~ 1.

Y +(IZ)

YM +(IZ)

As we have seen in Section 4.1.2, quantum coherence is entirely in the off-diagonal components of the density matrix. For example, in the case of a two-dimensional Hilbert space described by the basis formed by the two orthonormal states lSI} and IS2}, the most general density matrix has the form



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where I, ZY, and M represent the coinitiator, initiator, and monomer, respectively. Initiation by the combination of a Lewis acid and protogen or cationogen has the advantage over initiation by a Br nsted acid that the anion IZ is far less nucleophilic than A . This prolongs the life of the propagating carbocation and allows propagation to proceed to higher molecular weight. Initiation by the combination of an alkyl halide and Lewis acid can be achieved by using a hydrogen halide and a Lewis acid. The alkyl halide is produced in situ by addition of hydrogen halide to monomer, followed by reaction with the Lewis acid. There is evidence that Lewis acids initiate a slow polymerization in some (but not most) systems by a self-ionization process in addition to the coinitiation process [Balogh et al., 1994; Grattan and Plesch, 1980; Masure et al., 1978, 1980]. Two mechanisms are possible for self-initiation. One involves bimolecular ionization





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ISl}(SI!Pl + IS 2}(S2Ip2 + ISl}(s2la + IS2}(slla*,

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2AlBr3 AlBr+ (AlBr4) 2

(4.39)

followed by reaction with monomer:

and unless a =1= 0, Pwill have some quantum coherence. However, this identification of quantum coherence with the presence of off-diagonal components is possible only when the basis is orthogonal. In the example above, if we use the states l<Pl} and 1<P2} as basis instead of lSI} and IS2}, where

AlBr2M +(AlBr4)

(4.40)

The Lewis acid acts as both initiator and coinitiator. The second mechanism involves the addition of Lewis acid to monomer followed by reaction of the adduct with another molecule of Lewis acid:

and (4.41) then (4.42)

5-5

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[Le., two diagonal components with no off-diagonal components as if it is a statistical mixture even when Tr(p2) = ,I and it is a pure state]- all because I!f'l) and 1!f'2) are not orthogonal. The same trick can be played when the dimension of the Hilbert space is infinite, as in the case of a mode of the radiation field (a field oscillator). In this subsection we see that a basis of coherent states allows us to write down any density matrix as a diagonal matrix. To derive the diagonal coherent-state representation of the density matrix, we first notice that any density matrix p describing a single mode of the radiation field can be written as

Most of the evidence to support the self-ionization process is indirect, consisting of kinetic, conductance, and spectrophotometric data for polymerization at different levels of dryness and purity. One concludes that self-ionization occurs if polymerization is achieved in systems subjected to the most stringent puri cation and drying procedures. The major problem in ascertaining whether self-initiation occurs and, if it does, its extent relative to the initiation process involving a coinitiator is the large effect exerted by small amount of protogens or cationogens. Thus water concentrations of 10 3 M are suf cient to increase the initiation rate by a factor of 103 for TiCl4 and AlCl3 in CH2 Cl2 [Masure et al., 1978, 1980; Sauvet et al., 1978]. Polymerizations in the presence of sterically hindered pyridines such as 2,6-di-t-butylpyridine and 2,4,6-tri-t-butylpyridine offer further evidence for the presence of a self-ionization initiation process [Gandini and Martinez, 1988]. Sterically hindered pyridines (SHP) are generally active proton scavengers but do not react with Lewis acids or carbocations; in other words, steric hindrance cuts down on the reactivity of the nitrogen toward electrophilic species larger than protons. The presence of SHP results in complete inhibition of polymerization in some systems but only lowered reaction rates in other systems. For the latter, one observes a continuous decrease in polymerization rate with increasing concentration of SHP up to some critical SHP concentration. Thereafter, there is a residual (selfinitiation) polymerization rate that is unaffected by increasing SHP concentration. The

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