APPENDIX 1 Matrix Algebra of Spin-l/2 and Spin-l Operators It is frequently convenient to work with the matrix representation of spin operators in the eigenbase of the Zeeman Hamiltonian. Some results for spin-1/2 and spin-l systems are given in this Appendix. Eigenvectors Eigenvectors are represented as column matrices (kets) and row matrices (bras), while operators are square matrices. The Ill() and 1m states for spin-t/2 are represented by: Ill() == G) and 1{3) == G) (1) while, for spin 1: (2) 11) ~ m' 10) ~ m' and 1-1) ~ m Operators The matrix representation of spin operators in this eigenbase may be obtained by applying the results of angular momentum theory: IzII, m) = mil, m) I+II,m) = J(I - m)(I + m + 1) II,m + 1), 1+ = Ix + iIy (3) LII,m) = J(I + m)(I - m + 1)II,m -1), L = Ix - iIy J2 II,m) =1(1 + l)II,m). Operators 351 The matrix representation of the operator 12 is thus 1(1 + 1) times the unit matrix of dimension (21 + 1). The other operators are given by: Spin 1/2 Spin 1 (4) L G~) 1 0 0) 12 ~(l 0) 2 ( 0 1 0 4 0 1 001 The element ij of these matrices represents (iIOlj), Ii) being the ket corre­ sponding to m = mi (ml = I, m2 = I - 1, m3 = I - 2, ... ). The matrix representation of eigenvectors and operators in a composite spin system is obtained by working in the appropriate direct product space. Thus, for two spins 1/2, laa) ~ G)®G) ~ m (5) Ipa) ~ (~)®G) ~ m For such an IS spin system, we have in the direct product space: 352 Appendix 1. Matrix Algebra of Spin-lj2 and Spin-l Operators la) IP) (al (0 2Ix = (PI 1 ~ )@ ls= G ~)@(~ ~) laa) lap) IPa) IPP) (aal 0 1 (aPi 0 0 = (pal 0 0 (6) (PPI l! 0 ~J la) IP) 2S - 1 x (al ( 0 1) - C ~ x - I @ (PI 1 0 - 0 ~)@G ~) laa) lap) IPa) IPP} (aal 1 0 (aPI 0 0 (7) = (pal 0 0 (PPI ~~ 0 !J Note that the order of the spins is chosen arbitrarily; the choice made, however, must be maintained right through. The direct product of two ma­ trices A and B is given by: Ai:iB A12B ...... Ai:nBj C=A@B= .: .: (8) l . Ani B An2B ...... Ann B The direct product of an (m x m) matrix with an (n x n) matrix is thus an (mn x mn) matrix. Note that the direct product of matrices is therefore very different from matrix multiplication. Also listed below are the matrix representations of some higher powers of spin operators. These results may be checked by usual matrix multiplication. S~nlp S~nl ~ (H J ~ ( ~ ~ -~) -1 0 1 !(1 0) 4 0 1 (H ~) (9) Operators 353 Spin 1/2 Spin 1 0 [Ix, IyJ+ 0 0 i -i) (~ 0 -1 [Iy, IzJ + 0 0 fi (! -1 !) 1 [Iz,IxJ+ 0 1 (0 0 fl~ -1 -!) Various recursions then follow for i = x, y, z: If = iI7- 2 (I = 1/2, n ~ 2) (10) If = 17- 2 (I = 1, n ~ 3) For spin 1, the following relations also hold, with i = x, y, z: IJ)i =0 (i of. j) (11 ) IJf + If Ii = Ii (i of. j) In general, for a spin I, we have the Cayley identity: [Iz - IJ[Iz - (I - I)J[Iz - (I - 2)J ... [Iz + IJ =0 and I~I+l) = I~I+l) = 0; (12) /-i ) I~ ( . Il . (Iz - j) I~= 0, i = 1, 2, ... , I. J= -(/-,) Finally, we give below the matrix representation of various bilinear operator products for a two-spin-l/2 situation. The results may be verified by matrix multiplication, working with the representation of the individual single-spin operators in the direct product space of the two spins 1/2. 0 0 0 1 0 1 i [ 0 0 1 IxSx = 4 Y 1 0 IxS =4 0 -1 0 -~J ~~ 0 0 ~J 1 0 0 0 0 0 0 i 0 -1 1 0 1 IySx = 4 IySy = 4 0 -~1 0 -~J I~ 0 0 U0 0 354 Appendix 1. Matrix Algebra of Spin-1/2 and Spin-I Operators 0 0 1 0 1 0 -1 0 1 0 0 IzSx = (13) IzSz = 4 ~ 0 -1 4 0 0 ~ 0 0 ~J ~ 0 -1 -!J -1 0 0 1 i 1 0 0 1 0 0 IzSy = 4 ~o~ IxSz = 4 0 0 0 0 -] 0 -1 ~J l~ -1 0 0 -1 i 0 0 0 IySz = 4 ~ 0 0 [ -1 0 ~J From the above, it is clear that IxSx, IySy, IxSy, and IySx represent zero,- and double-quantum coherences in the spin-I/2 IS system. Operator representations for an IS system with I = 1, S = 1/2 involve (6 x 6) matrices. For example, 0 1 0) (0 1 j2Ix = (1 0 I ® Is = 1 0 o 1 0 0 I 11a) lIP) lOa) lOP) 1-1a) 1-1P) (lal 0 0 0 0 o (lPI 0 0 0 1 0 o (Oal 1 0 0 0 o (14) =(OPI 0 1 0 0 0 1 ( -lal 0 0 0 0 o ( -lPI 0 0 0 0 o (1 0 0) (15) 2Sx = II ® G~) ~ ~ ~ ~ 09 (~ ~) 0 0 0 0 1 0 0 0 0 0 0 0 0 (16) 0 0 1 0 0 0 0 ~1 0 0 0 0 0 0 0 1 ~j Operators 355 0 0 -1 0 0 0 0 0 0 -1 0 0 1 0 0 0 -1 0 j2Iy = i (17) 0 1 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 2Sy= i (18) 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1= (19) z 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 2Sz= (20) 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 APPENDIX 2 The Hausdorff Formula It is frequently necessary to evaluate expressions of the form: RA = exp( -iXt) A exp(iXt) which describe the evolution of an operator A for t seconds, under the action of a time-independent Hamiltonian x. The Baker-Campbell-Hausdorff for­ mula consolidates the series expansion of exponential operators to evaluate such operator evolutions: RA == exp (-iXt) A exp (iXt) (it)2 = A - (it) [X, A] + T![X, [X,A]] (1) (it)3 - 3! [X, [X, [X, A]]] + ... Note that this formula involves successive commutators with the Hamiltonian. An NMR problem has a well-defined, finite dimensional operator space associated with it and so the number of independent commutators is limited and the series expansion quickly leads to an operator recursion. It may be noted especially that it is crucial to work with the appropriate basis set of linearly independent operators, and to insure that successive commutators are expressed in this basis set, so that the operator recursions are not lost sight of. Suitable basis set operators for problems involving spin-l/2 and spin-l systems have been discussed in Chapter 1. We discuss below briefly some cases of interest. Appendix 2. The Hausdorff Formula 357 CASE 1. [£', A] = O. It follows that RA = A since all the commutators vanish. In other words, A is invariant to evolution under a Hamiltonian with which it commutes. For example consider a spin-1 system, with £' = tv; - 1(1 + 1)) and A = [/x,!y] + . [£', A] ~ [1;, Ixly + Iylx] = [1;, Ixly] + [1;, Iylx] = ilz(l; - In + i(l; - I;)Iz - iIAI; - I;) - iU; - I;)Iz (2) = 2i(lAI; - I;) + (I; - I;)Iz) = 2i(lz - Iz) = 0 where we have employed the properties of spin-1 operators discussed in Appendix 1, and some properties of commutators to be discussed later in this Appendix. Thus, RA = [1x'!y]+. CASE 2. [£', A] = aA. In this case, RA = A exp ( - iat). As an example, consider £' = M z and A = I + = Ix + ily • Then, [£', A] = iMy + Mx = L\A. Thus, RA = 1+ exp (-iL\t). CASE 3. [£', A] = B, [£', B] = kA, A =1= aB. This has been termed a problem of order 2. We find, RA = ( A + (i~{2 kA + (i~t k2A + ... ) _ (itB + (i~;3 kB + ... ) 2 4 2 3 _ A t k + t k _ ) _ iB (t _ t k + ) - (1 _ 2! 4! ... 3! ... (3) = A cos}kt - ~iB sin}kt Problems of order 2 have been encountered frequently in Chapters 1 and 3 and the corresponding equations of motion, R A , have been given there. We consider here a few representative examples. (1) A = Ix, £' = 2nJIzSz. [£', A] = i2nJIySz = B; (4) Thus, RA = Ix cos nJt + 2IySz sin nJt. (5) (2) A = Ix, £' = M z· [£',A] = iMy = B; (6) Thus RA = Ix cos M + Iy sin M. (7) 358 Appendix 2. The Hausdorff Formula Problems of order higher than two are encountered, for example, in AMXN systems evolving under a strong coupling Hamiltonian as discussed in Chapter 1 and lead to multiple frequencies of evolution. Commutator Algebra We give here some of the standard results of commutator algebra that are useful in evaluating the Hausdorff formula for specific Yf and A.