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where M0 and f1 0 are the initial values of M and f1 . Equation 6-15 allows the calculation of F1 as a function of f1 for a given set of r1 and r2 values. These can then be employed as F1 f1 to allow the graphical or numerical integration of Eq. 6-33 between the limits of f1 0 and f1 . In this manner one can obtain the



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The total average rate of variation of atomic energy is the sum of the individual rates (6.24) and (6.31). This sum becomes evident if we introduce a complete set of unperturbed atomic eigenstates in (6.24) and rewrite it as

variations in the feed and copolymer compositions with the degree of conversion (de ned as 1 M=M0 ). Equation 6-33 has been integrated to the useful closed form

(a Ifa(t)1 b) . (b Ifa(t)1 a)

1 M f1 1 M0 f1 0 !a f2 f2 0 !b ! f1 0 d g f1 d 6-34

(alra(t)lb) (b1fa(t)la}}.

which relates the degree of conversion to changes in the comonomer feed composition [Dionisio and O Driscoll, 1979; Meyer and Chan, 1968; Meyer and Lowry, 1965]. The zero subscripts indicate initial quantities and the other symbols are given by





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(6.32) Then the total rate is simply (6.33) As you can see from (6.33), when la} is the ground state, Wab will never be positive so that the summation will vanish. In other words, in the ground state the contribution of the vacuum fluctuations cancels the radiation reaction contribution exactly and the atom does not decay. This shows that the ground state of the atom is stable only because of the vacuum fluctuations of the field. The self-reaction term alone would lead to the collapse of the ground state as classical electrodynamics predicts and the atomic commutation relations [ri,pj] = ilioij would not hold. 4 So far, our treatment of spontaneous emission has been perturbative. We will now go beyond the perturbative approach and we will show that an initially excited atom decays exponentially to the ground state in free space. Before we do so, however, it is convenient to simplify the problem further.

6-35

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to classical theory, an atom cannot have a minimum-energy state even when only the Coulomb interaction is included in the Hamiltonian. It is the quantum commutation relations of the atomic operators that prevent the electron from colliding with the proton by giving it enough kinetic energy to overcome the Coulomb attraction every time the electron is localized within a certain neighborhood of the proton [409]. When the coupling with the radiation field is included to account for atomic emission, the atomic commutation relations are violated unless the field commutation relations are included [131, 556].

Equation 6-34 or its equivalent has been used to correlate the drift in the feed and copolymer compositions with conversion for a number of different copolymerization systems [Capek et al., 1983; O Driscoll et al., 1984; Stejskal et al., 1986; Teramachi et al., 1985]. The larger the difference in the r1 and r2 values of a comonomer pair, the greater is the variation in copolymer composition with conversion [Dadmun, 2001]. A few examples will illustrate the utility of Eqs. 6-33 and 6-34. Figure 6-3 shows the behavior observed in the radical copolymerization of styrene and methyl methacrylate. F1 and F2 are the instantaneous copolymer compositions for the instantaneous feed

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The canonical variables we have used to describe this system are not unique. They are related to other possible sets of canonical variables by unitary transformations; the quantum analog of classical contact transformations. We will perform such a transformation on these canonical variables to obtain a more convenient set of variables. When written in terms of this new set, the Hamiltonian will acquire a form more amenable to physical interpretation. We will then be able to use physical insight to make some approximations that will greatly simplify the problem. Let us consider the unitary transformation T that takes pinto p - (q/c)A (0), that is, TpTt = p- ~A(O). (6.34)

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