Journal of Matheatical Cheistry 24 (1998 123 132 123 On the suations involving Wigner rotation atrix eleents Shan-Tao Lai a, Pancracio Palting b, Ying-Nan Chiu b and Harris J. Silverstone c a Vitreous State Laboratory, The Catholic University of Aerica, Washington, DC 20064, USA b Center for Molecular Dynaics and Energy Transfer, Departent of Cheistry, The Catholic University of Aerica, Washington, DC 20064, USA c Departent of Cheistry, The Johns Hopkins University, Baltiore, MD 21218, USA Received 3 June 1997 Two new ethods to evaluate the sus over agnetic quantu nubers, together with Wigner rotation atrix eleents, are forulated. The first is the coupling ethod which akes use of the coupling of Wigner rotation atrix eleents. This ethod gives rise to a closed for for any kind of suation that involves a product of two Wigner rotation atrix eleents. The second ethod is the equivalent operator ethod, for which a closed for is also obtained and easily ipleented on the coputer. A few exaples are presented, and possible extensions are indicated. The forulae obtained are useful for the study of the angular distribution of the photofragents of diatoic and syetric-top olecules caused by electric-dipole, electric-quadrupole and two-photon radiative transitions. 1. Introduction Recently, we discussed [5,6] a nuber of suations involving Wigner rotation atrix eleents which are needed for the angular distribution of photofragents caused by dissociative electric-dipole, electric-quadrupole and two-photon radiative transitions. The basic ethod previously eployed involved the use of the recurrence relations of the Wigner rotation atrix eleents. In the following, we report two alternative ethods: (i the coupling operator ethod [4,7], and (ii the equivalent operator ethod. As illustrations, we shall use these ethods to perfor three kinds of suations. 2. The coupling ethod Fro the coupling rule of the Wigner rotation atrix eleents, we obtain the first kind of suation [4,7]: ( ( ( 3 d 0 (β 2 3 = ( P 0 3 (cos β, (1 J.C. Baltzer AG, Science Publishers
124 S.T. Lai et al. / Suations involving Wigner rotation atrix eleents where (... are Wigner 3 sybols, P (cos β is a Legendre polynoial, and 3 = 0, 1, 2,...,2. This closed for is useful in dealing with the following prototypical suation: k d (β 2, (2 for which k can assue the values 0, 1, 2,...,2. Fro equation (1 it is found that 2 d (β 2 = 1 3 ( + 1 3[ 1 ( + 1 3 2 ] P 2 (cos β, (2a 3 d (β 2 = 1 5 (32 + 3 1 cos β 1 5[ (3 2 + 3 1 5 2] P 3 (cos β, (2b 4 d (β 2 = 1 15 ( + 1(32 + 3 1 1 21 (62 + 6 5 [ ( + 1 3 2] P 2 (cos β + 1 [ 3( + 1( + 2( 1 5(6 2 + 6 5 2 + 35 4] 35 P 4 (cos β, (2c { 15 2 ( + 1 2 50( + 1 70( + 1 2 + 63 4 + 105 2 + 12 } d (β 2 = { 15 2 ( + 1 2 50( + 1 70( + 1 2 + 63 4 + 105 2 + 12 } P 5 (cos β (2d and { 5 3 ( + 1 3 + 105 2 ( + 1 2 2 315( + 1 4 + 231 6 + 40 2 ( + 1 2 525( + 1 2 + 735 4 60( + 1 + 294 2} d (β 2 = { 5 3 ( + 1 3 + 105 2 ( + 1 2 2 315( + 1 4 + 231 6 + 40 2 ( + 1 2 525( + 1 2 + 735 4 60( + 1 + 294 2} P 6 (cos β. Equations (2d and (2e can be reduced to the following closed fors: 5 d (β 2 = 1 {[ 15 2 ( + 1 2 50( + 1 + 12 ] 63 35 [ 2( + 1 3 ] 2 + 63 4} P 5 (cos β + 1 9[ 2( + 1 3 ]{ (3 2 + 3 1 P 1 (cos β (2e
S.T. Lai et al. / Suations involving Wigner rotation atrix eleents 125 [ (3 2 + 3 1 5 2] P 3 (cos β } 1 [ 15 2 ( + 1 2 50( + 1 + 12 ] P 1 (cos β (3a 63 and 6 d (β 2 = 1 21 ( + 1(34 + 6 3 3 + 1 1 [ ( + 1 3 2 ][ 5( + 1( 1( + 2 + 7 ] P 2 (cos β 21 + 1 [ ][ 3( + 1 7 3( + 1( 1( + 2 77 5(6 2 + 6 5 2 + 35 4] P 4 (cos β + 1 { 5( 2( 1( + 1( + 2( + 3 231 + 21 [ 5 2 ( + 1 2 25( + 1 + 14 ] 2 105 [ 3( + 1 7 ] 4 + 231 6} P 6 (cos β, respectively. The coupling rule ay also be used to obtain the second kind of suation: ( + k k ( 3 d +k 0 (βd k (β = ( ( + k k 3 0 (3b P 3 (cos β, (4 where 3 = 2k,2k + 1,...,2 and k = 0, 1/2, 1,...,. This equation can be reduced to [4,7] ( ( [( ( L + k 2 2][ ( k 2 2] 2k + 3 2 3 L + k + L L ( 3 2k d +k 3 L (βd k (β = ( L [ ( ( ( + k 2 2][ ( k 2 2] 2k + 3 L L ( 3 2k 3 L 2 3 + k + L P 3 (cos β, (5 where (... are binoial coefficients.
126 S.T. Lai et al. / Suations involving Wigner rotation atrix eleents A third kind of suation that can be obtained by this coupling ethod is ( + k ( 3 d +k 0 (βd (β = ( ( + k 3 0 P 3 (cos β. (6 In suary, the general coupling for is given by [4,2] ( 1 2 3 D 1 1 1 2 3 1 (αβγd 2 1 2 (αβγ 2 2 ( 1 2 3 = 1 2 D 3 3 3 (αβγ, (7 3 where D (αβγ ei α d (βeiγ. (8 3. The equivalent operator ethod We shall deal with the first kind of suation by using the so-called equivalent operator ethod. Fro the definition [3,5] d (β e iβj y, (9 we begin by deriving an equality. For two noncouting operators A and B, we have, by using the Baker Capbell Hausdorff identity, followed by the Taylor expansion, B = e iλa Be iλa = e iλa (iλ k B = k! Ak B, (10 in which A k B denotes the kth degree nested coutator, A k B = [ A, [ A,[A,...[A, B],...] ]]. (11 Under a siilarity transforation, A = A, and the operators B and B are said to be equivalent. For exaple, the Cartesian coponents of a rotational angular oentu obey the coutation relations [J x, J y ] = ij z, [J y, J z ] = ij x, [J z, J x ] = ij y. (12 Using equation (10, we have the equivalent-operator equality e iαjx J z e iαjx = J z cos α + J y sin α, (13 where α is the angle of rotation with respect to the x-axis.
S.T. Lai et al. / Suations involving Wigner rotation atrix eleents 127 Siilarly, we ay obtain a useful equivalent-operator equality e iβjy J z e iβjy = J z cos β J x sin β. (14 An alternative geoetrical interpretation of equations (10, (13 and (14 is given in the appendix. Here, the siilarity transforation of J z has been reduced to explicit fors which are ore convenient for operating upon rotational wave functions. We now give exaples of the usefulness of equation (14 for the evaluation of suations of the first kind. Exaple (1 d (β 2 = e iβj y J z e iβj y = e iβj y J z e iβjy = J z cos β J x sin β = cos β. (15 Here, we have used the resolution of the identity for the th irreducible vector space of O + (3, viz. = 1, (16a along with the identities J x = 1 2 (J + + J and J ± = 0. (16b Exaple (2 2 d (β 2 = (Jz cos β J x sin β 2 = 1 2 ( + 1 sin2 β + 2 (3 cos β 1. (17 2 Exaple (3 3 d (β 2 = (Jz cos β J x sin β 3 = 3 cos β + 2 sin2 β cos β [ 3( + 1 5 2 1 ]. (18
128 S.T. Lai et al. / Suations involving Wigner rotation atrix eleents Equations (15, (17 and (18 agree with the results obtained by using the recurrence relations [4,5,7]. We note that for the equivalent operator ethod the powers of the expression (J z cos β J x sin β are expressible as a syetric su, i.e., n (J z cos β J x sin β n = ( k{ (J z cos β n k (J x sin β k}, (19 where the brackets occurring in the right-hand side indicate the syetric su of the product contained therein. Notice that in the previous section we have considered the angular oentu operators and the functions, sin β and cos β to be independent of each other. Thus, they coute and we ay write equation (19 as n (J z cos β J x sin β n = (cos β k ( sin β n k{ Jx n k Jz k }. (20 For each given ter in the suation on the right-hand side there are n!/(k!(n k! ters that arise fro the expansion of the syetric su. It is convenient to have an expression for this expansion. By induction, in which { J n 1 x n 1 } J z = r 1 =0 n 2 { J n 2 x Jz 2 } ( = J r 1 r 1 =0. n k { J n k x Jz k } = r 1 =0 r 1 x J z J n 1 r 1 x, n 2 r 1 x J z r 1 x J z n k σ(k 1 r k =0 r 2 =0 n k σ(1 σ(k = r 2 =0 ( J r 2 x J z J n 2 r 1 r 2 x, r 2 x J z n k σ(2 r 3 =0 r 3 x J z k x J r z J n k σ(k x, (21 k r i. (22 In order to evaluate the various types of suations, we have need only of the diagonal parts of the syetric sus. Since J z is already a diagonal operator in the basis { }, it behooves us to consider the diagonal parts of powers of J x. For an operator Jx, p diagonal parts arise solely fro p = even integer = 2q, and we ay express the as 2q x diag = 1 { J q 2 2q + J } q, (23 i=1
S.T. Lai et al. / Suations involving Wigner rotation atrix eleents 129 by aking use of equation (16b. Thus, we are interested in the deterination of equation (20 in the for [ (Jz cos β J x sin β n] n diag = (cos β k ( sin β n k{ n k } = n x diag J k z ( (cos β k sin β n k {{ (n k/2 J 2 + J (n k/2 } J k z }, where the su is restricted to those values of k for which n k is an even integer. The syetric su ay be easily evaluated fro equation (21. The replaceents k = q, n = 2q, J x = J + and J z = J in that equation lead one to { J q + J q } q ( = J s 1 + J s 1 =0 q σ(k 1 s q=0 q σ(1 s 2 =0 s 2 + J q σ(2 s 3 =0 s 3 + J (24 ( s J q + J J q σ(k +. (25 To deterine the atrix eleents we ake use of the well-known stepping forulae J± k (!( ± + k! =, ± k. (26 ( ±!( k! First, we evaluate J q σ(k (![ + + q σ(q]! + =, + q σ(q, (27 ( +![ q + σ(q]! and then J+J s = ( + 1! ( +!( + + s 1! ( + 1! (!( s + 1!, + s 1. (28 We utilize these two forulae to obtain { J q + J q } q = s 1 =0 [ + σ(1]! [ + σ(1]! q σ(1 s 2 =0 [ + σ(2 1]! [ + σ(2 + 1]! [ + σ(1 + 1]!( +! [ + σ(1 1]!(! [ + σ(2 + 2]![ + σ(1 + 1]! [ + σ(2 2]!( + σ(1 1]!
130 S.T. Lai et al. / Suations involving Wigner rotation atrix eleents.. q σ(l 1 s l =0 [ + σ(l l + 1]! [ + σ(l + l 1]! [ + σ(l + l]![ + σ(l 1 + l 1]! [ + σ(l l]!( + σ(l 1 l + 1]! q σ(q 1 s q=0 [ + σ(q + q ][ + σ(q q + 1 ] (![ + σ(q 1 + q 1]! ( +!( + σ(q 1 q + 1]!, (29 where the σ s are given by equation (22. The usage of this result is straightforward. It is, of course, valid for q>0. For q = 0, the syetric su {J 0 + J 0 }issiply1. With equation (29 in hand, the desired atrix eleents of the diagonal parts, [(J z cos β J x sin β n ] diag, are quite siple to evaluate, since all operators in equation (24 are diagonal and, therefore, coute. Thus, we ay write it as (Jz cos β J x sin β n = n ( ( cos β k sin β n k { (n k/2 J 2 + J (n k/2 }. (30 The advantage of having a closed analytical for for this expression is apparent: it is readily ipleented on the coputer. Finally, it should be noted that, since Wigner rotation eleents are related to the spherical haronics by D 0 (αβγ = 4π 2 + 1 Y (βα = ( 4π 2 + 1 Y (βα, (31 all of the suations over d (β can be easily reduced to suations over spherical haronics Y (βα by setting = 0. Appendix The geoetric view of equations (10, (13 and (14 can be discussed as follows. Using the convention of a right-handed syste of coordinates, let us consider rotating a given position vector r of coponents (x, y, z about an axis n by angle φ, thereby yielding r (cf. figure 1. That is, r = R(φ, nr, wheren akes an angle θ with r. For the convenience of discussion, we copare the scalar coponents with their vector
S.T. Lai et al. / Suations involving Wigner rotation atrix eleents 131 Figure 1. directions after the rotation. To visualize the relative positions of r and r,wetake n to be the z-axis and the y-axis to be the coon plane of n and r before rotation through angle φ. Thex-axis will be perpendicular to n and r (and y. For vector r : scalar coponents vector direction x = r sin θ cos φ, r n, y = r sin θ cos φ, r (r nn, z = r cos θ, (r nn, and 0a = r sin θ, r n = 0a sin φ, r (r nn = 0a cos φ. Therefore, we obtain 0a in ters of r n and r (r nn, i.e., Finally, we have 0a = (r nsinφ + ( r (r nn cos φ. r = R(φ, nr = (r nn + (n rsinφ + ( r (r nn cos φ. (A.1 This result is obtained by rotating the vector r. If the rotation were effected upon the coordinates (i.e., φ is replaced by φ, the result would be r = (r nn (n rsinφ + ( r (r nn cos φ. (A.2
132 S.T. Lai et al. / Suations involving Wigner rotation atrix eleents Since the rotational angular oentu transfors as a vector under rotation, we, therefore, have the following siilarity transforation upon replacing the vector r by J: J = e iφ n J iφ n J J e = (J nn (n Jsinφ + (J nn cos φ. (A.3 This rotation of the coordinate syste follows the convention of Rose [8], Brink and Satchler [2] and Biedenharn and Louck [1]. In contrast, J = e iφ n J iφ n J J e = (J nn + (n Jsinφ + (J nn cos φ (A.4 represents a rotation of the physical syste. For exaple, e iφjz J y e iφjz = J y cos φ + J x sin φ, (A.5 where n J = J n = J z and J x = J y = 0. Edonds [3] uses this latter convention. References [1] L.C. Biedenharn and J.D. Louck, Angular Moentu in Quantu Physics, Theory and Applications, Encyclopedia of Matheatics and its Applications, Vol. 8 (Addison-Wesley, Reading, MA, 1981. [2] D.M. Brink and G.R. Satchler, Angular Moentu (Oxford University Press, Oxford, 1962. [3] A.R. Edonds, Angular Moentu in Quantu Mechanics (Princeton University Press, Princeton, NJ, 1960. [4] S.T. Lai, Ph.D. thesis, The Catholic University of Aerica (1991. [5] S.T. Lai and Y.N. Chiu, J. Math. Phys. 30 (1989 844. [6] S.T. Lai and Y.N. Chiu, J. Math. Phys. 31 (1990 1261. [7] S.T. Lai, P. Palting and Y.N. Chiu, J. Mol. Sci. (Beiing 10 (1994 1. [8] M.E. Rose, Eleentary Theory of Angular Moentu (Wiley, New York, 1957.