QUESTION: An evaluated _r

Since _r is a scalar product of two first-rank tensors, r and , we can use (6.21) below to determine the matrix elements of _r.

Show that, it can yield this result

or

since _r has odd parity, it follows that

so that . Thus,

~

*additional notes that may be handy

The C-coefficient and Racah coefficient in (6.21) are obtained from tables I.2 and I.3, respectively.

- (, .x, x x,, (j'm'jj2|T_(1).T,(2)|jm|1j2) = Omm jri(-Jitjr- WG1j2hz; jL) x [(25 + 1)(25 + 1)]* (6 || Tz(1) || 31)62 || T-(2) || 12) (6.21) M'EM j' : 1x1 - x) X x':-* - el-x migo 1+1 Table 1.2. Cilj; m-me, m) mg = 1 Q1+G++1) G1--*+1)G1+m+1) (2+1)(2+2) (2+1)61+1) Gi+m)Gi-m+1) 201611) T+1) G1--G--+1) G-m) + m) 21(271+1) (2011) mg-1 Gi-mG-2 (25+1)C2+3 [On-ne)(1+m+1) 27161 +1) 1 1-1 - [9 19 Gm +16+) 27.(2011) Table 1.3. Wabcd; if) Q = +1 (6+++)(+d-+1) (+)+b-d) (26+-1)(26--2)(22-+-1)(22+2)] (-)6+1-72b(25+1) (22 +1)(2d+2) c-d-1 (-)0+0-1 [U+6d+1)06+d) (-)+-1-1 (6+d+/+1)(6+d-1 26(26+-1)2(20+1) cmd+} (-1009 [ ) - [ No entant et),]" 1" - (, .x, x x,, (j'm'jj2|T_(1).T,(2)|jm|1j2) = Omm jri(-Jitjr- WG1j2hz; jL) x [(25 + 1)(25 + 1)]* (6 || Tz(1) || 31)62 || T-(2) || 12) (6.21) M'EM j' : 1x1 - x) X x':-* - el-x migo 1+1 Table 1.2. Cilj; m-me, m) mg = 1 Q1+G++1) G1--*+1)G1+m+1) (2+1)(2+2) (2+1)61+1) Gi+m)Gi-m+1) 201611) T+1) G1--G--+1) G-m) + m) 21(271+1) (2011) mg-1 Gi-mG-2 (25+1)C2+3 [On-ne)(1+m+1) 27161 +1) 1 1-1 - [9 19 Gm +16+) 27.(2011) Table 1.3. Wabcd; if) Q = +1 (6+++)(+d-+1) (+)+b-d) (26+-1)(26--2)(22-+-1)(22+2)] (-)6+1-72b(25+1) (22 +1)(2d+2) c-d-1 (-)0+0-1 [U+6d+1)06+d) (-)+-1-1 (6+d+/+1)(6+d-1 26(26+-1)2(20+1) cmd+} (-1009 [ ) - [ No entant et),]" 1