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is a solution of (4.71). As any function of Ii; commutes with X, f (x) would be completely arbitrary if all we had was (4.71). There is, however, another key equation that Z must satisfy.

C(CH3)2



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SSee VI of [243]. 6The reader who is not familiar with this should have a look at VI of [243]. 7Notice that the coefficients of xp and px in the sum have to be the same; otherwise. the sum is not Hermitian. This is the reason for adopting the symmetric product (xp + fd)/2 rather than just any arbitrary linear combination of xp and px.

Select the text that contains the formatting that you want to copy. Click the Home tab on the Ribbon. Click the Format Painter button ( ).

n CH2CH(CH3)2

As th. is unitary, it must preserve the commutation relations between operators; in particular, it cannot alter (4.60). In order not to alter (4.60), we must have

CH2(BF3OH)

(4.73)

+ H CH2C(CH3)2

(4.74)

5-18





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The only way to satisfy both (4.71) and (4.74) is to take f(i:) as a complex number. From (4.67) we see that a complex f(i:) just leads to a trivial phase factor in S>.. But we have already chosen the phase of 'I/J>.(x) when we wrote (4.63), and to satisfy (4.63), f (i:) = 0, so Z = -2 xp+px . I(AA AA) (4.75) We have determined the operator that performs an infinitesimal squeezing operation. Finite squeezing can be achieved by a succession of several infinitesimal squeezing operations. In other words, as S>. has the group property (4.76) we can obtain S>. from the product of N infinitesimal operators SHe if we choose N large enough so that (1 + )N = A. Then let us take = (lnA)/N, so that as N -+ 00, limN--+oo(1 + )N = limN--+oo[1 + (lnA)/N1N = exp(lnA) = A. Using (4.76), we find that

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This reaction may account in part for the oligomers obtained in the polymerization of propene, 1-butene, and other 1-alkenes where the propagation reaction is not highly favorable (due to the low stability of the propagating carbocation). Unreactive 1-alkenes and 2-alkenes have been used to control polymer molecular weight in cationic polymerization of reactive monomers, presumably by hydride transfer to the unreactive monomer. The importance of hydride ion transfer from monomer is not established for the more reactive monomers. For example, hydride transfer by monomer is less likely a mode of chain termination compared to proton transfer to monomer for isobutylene polymerization since the tertiary carbocation formed by proton transfer is more stable than the allyl carbocation formed by hydride transfer. Similar considerations apply to the polymerizations of other reactive monomers. Hydride transfer is not a possibility for those monomers without easily transferable hydrogens, such as N-vinylcarbazole, styrene, vinyl ethers, and coumarone. The two types of chain transfer to monomer are kinetically indistinguishable, but one (Eq. 5-17) results in unsaturated end groups, while the other (Eq. 5-18) results in saturated end groups. Chain transfer to counterion, also called spontaneous termination, involves transfer of a b-proton to the counterion. The initiator coinitiator is regenerated by its expulsion from the propagating species and, as in chain transfer to monomer, the polymer molecule has a terminal double bond

[~(xp+pi:)lnA]

H CH2C(CH3)2

n CH2C(CH3)2(BF3OH)

[~(a2 - at2 ) InA] .

Click and drag over the text to which you want to apply the same formatting. Word immediately copies the formatting to the new text.

BF3 OH2 + H CH2C(CH3)2

(4.77)

n CH2C(CH3)

Let us examine now a particular type of squeezed state that is also a minimum uncertainty state. This type of squeezed state can be seen as a generalization of coherent states where the product of uncertainties in the two canonically conjugate quadratures is still minimum, but with each quadrature having a different uncertainty8 (Le., variance). In fact, as the squeezing operation does not alter the uncertainty product, we can obtain such a state in a straightforward way just by applying S>. to a coherent state. But we can get more insight if instead of doing so, we calculate A At 8>.0.8>.. From (4.58), (4.59), (4.70), and (4.73), taking A -+ 1/ A, we find that (4.78)

or, in more general terms,

b).. =

HMnM +(IZ)

j-t).. v)..

Mn + 1 + H +(IZ)

+ v)..a t ,

Chain transfer to counterion differs kinetically from chain transfer to monomer in that the rate of chain transfer to monomer has a rst-order dependence on monomer while chain

(4.79)

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