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where Enml is the Levy-Civita symbol and 8n ,m is the Kronecker delta. 2. Two biquaternionsZ = zo+zlil +z2i2+Z3i3 andZ' ~ zb+z~il +Z~i2+Z~i3 are equal only if their components are equal (Le., Zo = zb, ZI = z~. Z2 = z~, and Z3 = z~). 3. The quaternion conjugate of Z Zo - Zlil - Z2i2 - Z3i3.



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The Diels Alder reaction, involving the [4 2]-cycloaddition of an unsaturated group (dienophile) to a 1,3-diene, has been studied for the synthesis of ladder polymers, such as the reaction of 2-vinyl-1,3-butadiene with benzoquinone [Bailey, 1972] (Eq. 2-247). Related polymerizations are those utilizing the [2 2]-cycloaddition reaction [Dilling, 1983]. While [4 2]-cycloaddition reactions are thermally induced, [2 2]-cycloaddition reactions are

6. The Hermitian conjugate, or as Hamilton called it, the biconjugate, of Z = Zo + z Ii 1 + Z2i2 + Z3i3, is given by zt = Z* = Zo - Ziil - Z2i2 - Z;i3. 7. The scalar part of a biquaternion (or quaternion) Z is given by 8(Z) = (Z + Z) /2. and for a product .z1~' 8 (ZIZ2) = 8 (Z2Zll. In Lanczos's method, the four-momentum becomes a biquaternion and so does the wavefunction. Let us try to factorize (3.24) as Dirac did but using biquaternions





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2-247

16Hamilton called them i, j, and k, but we will use h, h, and h instead to avoid confusing the first generalized imaginary unit with the old commutative i. 17Ifthe components of Z are not real, the quaternion modulus can vanish, be negative, oreven, be complex.

photochemically induced. The self-addition of alkene double bonds is the most extensively studied [2 2]-cycloaddition reaction, such as the polymerization of 2,5-distyrylpyrazine

To clarify this point, let us consider a specific example. Let us use the canonical quantization procedure to derive the wavefunction quantum mechanics for a particle of mass m moving in a potential V (x) in one dimension. The quantum Hamiltonian is then the same as the classical one, but with the classical momentum p and position x replaced by the operators6 p and X,

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(Eq. 2-248) [Braun and Wegner, 1983; Hasegawa et al., 1988, 1998]. This polymerization is a solid-state reaction involving irradiation of crystalline monomer with ultraviolet or ionizing radiation. The reaction is a topochemical or lattice-controlled polymerization in which reaction proceeds either inside the monomer crystal or at defect sites where the product structure and symmetry are controlled by the packing of monomer in the lattice or at defect sites, respectively.

From equation (2.1) we find that the Poisson bracket of x and pis (x,p) commutator of x and p must be

2-248

(2.17)

Polymer chains based on spiro structures have been studied as another route to heat-resistant polymers [Kurita et al., 1979]. A spiro structure is a double-strand structure (Sec. 1-2c) in which the uninterrupted sequence of rings have one atom in common between adjacent rings. (Adjacent rings in ladder polymers have two or more atoms in common.) An example of a spiro polymer is the polyspiroketal synthesized from 1,4-cyclohexanedione and pentaerythritol (Eq. 2-249). The low solubility and intractability of spiro polymers makes it dif cult to synthesize or utilize high-molecular-weight polymers.

1. So the (2.18)

2-249

If you want to index text that spans a range of pages, select the text and create a bookmark for it (click the Insert tab and click Bookmark). In the Mark Index Entry dialog box, click the Page range radio button. Click the Bookmark and choose the bookmark you just created, click Mark and click Close.

Pseudopolyrotoxanes and polyrotoxanes are polymers composed of a linear polymer chain threaded through one or more cyclic species [Gibson and Mahan, 2000; Raymo and Stoddart, 1999]. Synthesis of pseudopolyrotoxanes and polyrotoxanes involves polymerization of monomer(s) in the presence of the cyclic species. Cyclodextrins, crown ethers, paraquots, and other macrocyclics are used as the cyclic component. Both step and chain polymerizations have been successfully used.

[x,p]

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