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(4.89) where k~ == n7r/Lf. Even though the expression for the fields in terms of annihilation and creation operators is different before and after the sudden change in cavity length, the field operators, like q and p for the harmonic oscillator, remain unchanged. 12 Using this, we can work out how the annihilation and creation operators after the sudden change relate to the annihilation and creation operators before. Assuming that L f :::; L i , we multiply the electric field expressions in (4.88) and (4.89) by J2/Lfsin(n7rx/Lf) and integrate over x from 0 to Lf. Equating the results, we find that



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where YM IZ is the total concentration of all-sized propagating centers:

= __

YM IZ Kki I ZY M kt 5-33

t -bm )

Excel removes the cells and their content from the worksheet. Other cells shift over or up to fill the void of any cells that you remove from your worksheet.

Rp Ri kp M Kki kp I ZY M 2 kt kt 5-34

([if - ~]

The number-average degree of polymerization is obtained as the propagation rate divided by the termination rate:

Lf )

Xn Rp kp M kt Rt 5-35

(4.90)





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When chain breaking involves chain transfer to monomer (Eqs. 5-18 and 5-19), spontaneous termination (Eq. 5-20), and chain transfer to chain-transfer agent S in addition to combination with the counterion, the concentration of the propagating species remains unchanged (assuming relatively small amounts of S such that the coinitiator is not inactivated), and the polymerization rate is again given by Eq. 5-34. However, the degree of polymerization is decreased by these other chain-breaking reactions and is given by the polymerization rate divided by the sum of all chain-breaking reactions:

Analogously, multiplying the magnetic field expressions in (4.88) and (4.89) by J2/ L f cos(n7rx/ Lf), integrating over x from 0 to Lf, and equating the results, we obtain

Xn Rp Rt Rts Rtr;M Rtr;S 5-36

Rts kts YM IZ Rtr;M ktr;M YM IZ M Rtr;S ktr;S YM IZ S 5-37 5-38 5-39

(4.91)

Xn kp M kt kts ktr;M M ktr;S S 5-40

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(4.92) As the quantum state of the field remains unchanged for a sudden change in the Hamiltonian, (4.92) implies that the initial vacuum becomes squeezed. But as we had already foreseen, the new annihilation operators are related not only to the annihilation and creation operators of the corresponding oscillator (mode) before the sudden change, but also to those of every other oscillator (mode) as well.

To delete a whole column or row, click the column or row heading to select it; then click the Delete button in the Home tab. Excel deletes any existing data within the selected column or row and moves subsequent columns or rows to fill the space left by the deletion. To remove a cell s formatting without removing the content, select the cell you want to edit, click the Home tab, click the Clear button ( ) and choose Clear Formats to remove the cell s formatting.

1 kt kts S CM CS M X n kp M kp M 5-41

Any reader who has ever browsed through the literature on squeezing has probably come across pictures of noise ellipses and circles. In this subsection we explain how this geometrical picture of quantum noise works. In his pioneering paper on squeezed states, Yuen [660] defined his two-photon coherent states as in (4.79): (4.93) except that rather than demanding real JL and

where CM and CS are the chain-transfer constants for monomer and chain-transfer agent S de ned by ktr;M =kp and ktr;S =kp , respectively. Equation 5-39 is the cationic polymerization equivalent of the previously described Mayo equation (Eq. 3-108) for radical polymerization. For the case where chain transfer to S (Eq. 5-29) terminates the kinetic chain, the polymerization rate is decreased and is given by

his only requirement on them was (4.94)

Rp Kki kp I ZY M 2 kt ktr;S S 5-42

otherwise leaving them free to assume any complex value. Apart from a global arbitrary phase multiplying bl',v, Yuen's requirement on JL and II allows us to write

The various rate expressions were derived on the assumption that the rate-determining step in the initiation process is Reaction 5-4. If this is not the situation, the forward reaction in Eq. 5-3 is rate-determining. The initiation rate becomes independent of monomer concentration and is expressed by

= e-iO coshr,

Ri kl I ZY 5-43

e sinh r,

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