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Let us denote by O j = End(H j ) the space of linear operators (endomorphisms) acting on H j This is a complex vector space of dimension (2 j + 1)2 and an algebra for the natural composition of endomorphisms The spin spherical harmonic basis allows us to write a linear endomorphism of H j (i e an element of O j ) in a matrix form This provides the algebra isomorphism ~ O j = M(2 j + 1, C) = M 2 j+1 , the algebra of complex matrices of order 2 j + 1, equipped with the matrix product The projector |x x| is a particular linear endomorphism of H j , that is, an element of O j Being Hermitian by construction, it may be seen as a Hermitian matrix of order 2 j + 1, that is, an element of Herm2 j+1 M 2 j+1 Note that Herm2 j+1 and M 2 j+1 have respective (complex) dimensions ( j + 1)(2 j + 1) and (2 j + 1)2



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According to our now-familiar prescription, the coherent state quantization associates with the classical observable f : S 2 C the quantum observable f Af =

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A[n]: DA: HEL: In: SA: SL: lPv6 address Destination address Header extension length Intermediate Node n Source address Segments left

(173)

545 550 522

This operator is an element of O j Of course its existence is subjected to the convergence of (173) in the weak sense as an operator integral The expression above gives directly its expression as a matrix in the spin spherical harmonic basis, with matrix elements (A f ) :





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ar creates, modifies, or extracts archives. It is most commonly used to create static libraries, which are files that contain one or more object files. ar also creates and maintains a table that cross-references symbol names to the members in which they are defined. The ar command has the following syntax: ar {dmpqrtx} [options] [member] archive file [...] ar creates the archive named archive from the file(s) listed in file. At least one of d, m, p, q, r, t, and x is required. You will usually use r. Table 29-4 lists the most commonly used ar options.

S2 j (x) Y j (x)

with (A f ) =

200 0

(174)

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When f is real-valued, the corresponding matrix belongs to Herm(2 j+1) Also, we have A = (A f ) (matrix transconjugate), where we have used the same notation f for the operator and the associated matrix Note the interesting expression of the lower symbol of A f , easily obtained from the expression (681) of the overlap of two sigma-spin coherent states, and viewed

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f ( ) r (175)

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Also note that the map (173) can be extended to a class of distributions on the sphere, in the spirit of 12

The quantization of an arbitrary spin harmonic Y kn yields an operator in H j whose (2 j + 1) ~ (2 j + 1) matrix elements are given by the following integral resulting from (173): A Y kn

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Figure 4.8-13 Absorbance differences AII during electrochemical oxidation of poly(methy1thiophene) in an organic electrolyte (a), measured in situ, according to Neugebauer et al., 1983, and corresponding values for the oxidation of poly(ani1ine) in an acidic electrolyte (b), according to Kuzmany et al., 1988. The indicated potential is vs. SCE.

(x) Y kn (x) (dx) (x) Y kn (x) (dx) (176)

As asserted in Appendix C (Section C7), it is only when + = 0, that is, when = 0, that the integral (176) is given in terms of a product of two 3 j symbols Therefore, the matrix elements of AY m in the spin spherical harmonic basis are given in terms of the 3 j symbols by AY = ( 1) (2 j + 1) (2 + 1) 4 j j j j (177)

In the simplest case j = 1, we nd for the matrix elements (177) AY 10 AY 11 = = 3 1 m mn , 4 j ( j + 1) 3 1 4 j( j + 1) ( j n)( j + n + 1) mn+1 , 2 (178) (179)

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