Wigner 3-j Symbols. D j 2. m 2,m 2 (ˆn, θ)(j, m j 1,m 1; j 2,m 2). (7)

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1 Physics G6037 Professor Christ 2/04/2007 Wigner 3-j Symbols Begin by considering states on which two angular momentum operators J and J 2 are defined:,m ;,m 2. As the labels suggest, these states are eigenstates of J i 2 and J i z : J i 2,m ;,m 2 = h 2 j i j i +,m ;,m 2 Jz j i,m ;,m 2 = hm i,m ;,m 2 2 for i =, 2. We next introduce the Clebsch-Gordan coefficients that determine how these states can be combined to produce states which are eigenstates of J 2 and J z where J = J + J 2 and J 2 j, m;, = h 2 jj + j, m;, 3 J z j, m;, = hm i j, m;, 4 These coefficients are defined by: j, m,m ;,m 2 = j, m;,,m ;,m 2. 5 Define a general rotation acting on such a J 2, J z eigenstate to take the form: j exp i J ˆnθ/ h j, m = D j m,m ˆn, θ j, m 6 m = j If we insert the product of operators exp+i J ˆnθ/ h exp i J ˆnθ/ h in the matrix element in Eq. 5 we find that the Clebsch-Gordan coefficients obey the following relation: j, m,m ;,m 2 = j m = j m = m 2 = D j m,m ˆn, θ D m,m ˆn, θ D m 2,m 2 ˆn, θj, m,m ;,m 2. 7 We can remove the complex conjugate on the first D j m,m ˆn, θ factor by recognizing that in our usual conventions: e i J ˆnθ/ h = e i Jxnx+Jyny Jznzθ/ h = e +iπjy/ h e i J ˆnθ/ h e iπjy/ h. 8

2 By commuting e iπjy/ h with J z and J x ± J y it is easy to show: e iπjy/ h m,m = j m δ m, m. 9 Here the overall constant phase j can be deduced by applying the operator exp iπj y / h to an example of the state j, +j constructed from the product of 2j spin-/2 states, each with J z =+ h/2. We can now substitute Eq. 8 into Eq. 7 to replace D j m,mˆn, θ by 2j m m D j m, mˆn, θ. Finally the factors m and m and the reversed signs of m and m, can be absorbed by defining new coefficients which then obey a symmetrical transformation law. Thus, define the Wigner 3-j symbols: = m j, m j m m m,m ;,m j + These new symbols now obey: j = m m m 2 m = j m = m 2 = D j m,m ˆn, θd m,m ˆn, θ D m 2,m 2 ˆn, θ m m m 2. The transformation properties given in Eq. then imply that we can use these Wigner 3-j symbols to combine a collection of states made of the product of three representations of the rotation group into a state with total angular momentum zero. That is, the state j 3 m = m 2 = m 3 = j 3 m m m 2,m ;,m 2 ; j 3,m 3. 2 is not changed when acted on by a general rotation and hence must have J 2 = 0 and J 3 = 0. Since there is at most a single way to combine three angular momenta to form a state with J 2 = 0 and J 3 = 0, this implies that the Wigner 3-j symbols have a unique dependence on the three variables m, m 2 and m 3. Since the transformation rules given in Eq. are symmetrical with respect to the interchange of any j a,m a j b,m b pair, the uniqueness of the 2

3 Wigner 3-j symbols implies that they must be similarly symmetrical: j j 3 m m 2 m 3 = ca, b, c ja j b j c m a m b m c. 3 Given the normalization implied by Eq. 0, the Wigner 3-j symbols obey the condition: j 3 2 j j 4 =. 4 m m 3 m 4 m = m 2 = m 3 = j 3 The factor of / 2j + in Eq. 0 implies that the sum over m 2 and m 3 in Eq. 4 should return divided by the square of this extra factor. The final sum over m then gives 2 +/2 + =. Squaring and summing both sides of Eq. 3 over m, m 2 and m 3 then gives: ca, b, c 2 = 5 implying that ca, b, c is a simple phase. In fact, with our choices this phase is ca, b, c=ɛ abc + +j 3. 6 To establish Eq. 6 we must first adopt sign conventions for our Clebsch- Gordan coefficients. This contains two parts. The first is to determine how our Clebsch-Gordan coefficients change when we reverse the order of the two spins being coupled. Here we adopt the standard conventions: j, m,m,,m 2 = j j, m,m 2,,m. 7 For the case, the two quantities related by Eq. 7 are quite separate and this is simply a convention that can be imposed. For example, for the case > we could adopt any convention we wanted for j, m,m,,m 2 and then view Eq. 7 as the definition of j, m,m,,m 2 for the case >. However, when = this is a real condition and must be verified. This is not difficult to do if we consider the exchange operator S acting on the composite state,m ;,m 2 : S,m ;,m 2 =,m 2 ;,m. 8 For the case with largest value of m + m 2 =2, there is only a single state and it has eigenvalue for S. Thus, j, m,m,,m 2 must be symmetric. 3

4 Since for this case j =2, this agrees with the factor j =. When we consider the next lowest value of m + m 2 =2 there are two states:, ;, and, ;,. 9 From these we can form two eigenstates of S with eigenvalues ±. Since the lowered 2, 2 ;, necessarily has S = + since the lowering operation commutes with S, the new orthogonal state 2, 2 ;, must necessarily have S =. This implies that 2,m,m,,m 2 must be anti-symmetric under the exchange of m and m 2, again consistent with Eq. 7. When we lower again, we must deal with three states:, ;, 2,, ;, and, 2;,. 20 These can be written as two states that have S = + and one state with S =. The lowered versions of the two states we have already constructed have S = ±. Thus, the new state with j =2 2 must have S = +. This implies a symmetric Clebsch-Gordan coefficient 2 2,m +m 2,m,,m 2, again consistent with Eq. 7. It should now be clear how this pattern will continue, establishing the condition Eq. 7 for arbitrary 0 j 2. The combination of Eq. 7 and the factor j 3 in the definition, Eq. 0, then establishes Eq. 6 for the case that the second and third columns are being exchanged: a =,b =3,c = 2. The factor j 3 will introduce a phase 2 j 3 under such an interchange. This combines with the added factor j 3 coming from Eq. 7 with the appropriate change of variables to give: 2 j 3 j 3 = + 3j 3 = + +j 3 2 since 4j 3 is necessarily an even integer. The second case we must consider is the exchange of the first and second columns, 2 in Eq. 3 or a =2,b =. For this operation to be determined we must impose a further sign convention on our Clebsch-Gordan coefficients: j,, ;, > We will determine the phase c2,, 3 by comparing the sign of two quantities related by this phase: j 3 j2 j = c2,, 3 j j 3 j 3 j 3 j 3 4

5 From the condition in Eq. 22 and the definition of the Wigner 3-j symbols in Eq. 0 we learn that the sign of the left-hand-side of Eq. 23 is j 3 j 3 = 2 j The sign of the Wigner 3-j symbol appearing on the right hand side of Eq. 23 is the sign of the quantity: j 3,,j 3 ; j 3, j The conventions imposed in Eq. 22 imply, j 3, ; j 3, j 3 > We can then use the lowering operator + j 3 > 0 times to equate the positive sign of this quantity with the sign of the Clebsch-Gordan coefficient appearing in Eq. 25. Thus, equating the signs on the left and right hand sides of Eq. 23 we find: 2 j 3 = c2,, 3 j 3 27 or c2,, 3= +3 j 3 = + +j 3 as claimed. 5