Consistent chiral three-nucleon interactions in ... - Theory Center

3 Mathematical basics
the original Jacobi coordinates by using a HOB
|(n12l12( ξ ′ 1 ), n3l3( ξ ′ 2 ))LML〉
=
ñ12˜l12 ñ3˜l3 〈〈ñ12 ˜ l12, ñ3 ˜ l3; L|n12l12, n3l3〉〉1
3
(3.115)
×δ 2ñ12+ ˜ l12+2ñ3+ ˜ l3,2n12+l12+2n3+l3 |(ñ12 ˜ l12( ξ1), ñ3 ˜ l3( ξ2))LML〉 .
The spatial matrix element of the transposition operator follows as
〈(n ′ 12 l′ 12 , n′ 3 l′ 3 )L′ M ′ L |T23|(n12l12, n3l3)LML〉
= δL ′ ,L δM ′ L ,ML δ2n ′ 12 +l′ 12 +2n′ 3 +l′ 3 ,2n12+l12+2n3+l3 〈〈n′ 12l ′ 12, n ′ 3l ′ 3; L|n12l12, n3l3〉〉1
3
(3.116)
Combining the results in eq. (3.108), (3.109) and (3.116) leads to the total matrix
element of the transposition operator
〈[(n ′ 12 l′ 12 , s′ ab )j′ 12 , (n′ 3 l′ 3 , sc)j ′ 3 ]J ′ M ′ J , (t′ ab tc)T ′ M ′ T |
=
L ′ ,S ′ M ′ L ,M′ L,S ML,MS
S
⎧ ⎫⎧
⎪⎨ l12 sab j12⎪⎬
⎪⎨
× l3 sc j3
⎪⎩ ⎪⎭ ⎪⎩
L S J
× T23|[(n12l12, sab)j12, (n3l3, sc)j3]JMJ, (tabtc)TMT 〉
ˆj ′ 12 ˆj ′ 3 ˆ L ′ ˆ S ′ˆj12 ˆj3 ˆ L ˆ Sˆs ′ ab ˆsab ˆt ′ ab ˆtab
l ′ 12 s ′ ab j′ 12
l ′ 3 sc j ′ 3
L ′ S ′ J ′
× (−1) 1+t′ ab +tab (−1) 1+s ′ ab +sab
⎫
⎪⎬
L S
J
⎪⎭ ML MS
MJ
ta tb t ′
ab sa sb s ′
ab
tc T tab
× δT,T ′ δMT ,M ′ T δS,S ′ δMS,M ′ S δL ′ ,L δM ′ L ,ML
sc S sab
L ′ S ′
M ′ L M ′ S
J
′
× δ2n12+ ′ l ′ 12 +2n′ 3 +l′ 3 ,2n12+l12+2n3+l3 〈〈n′ 12l ′ 12, n ′ 3l ′ 3; L|n12l12, n3l3〉〉1 . (3.117)
3
Now we can eliminate the summations over L ′ , M ′ L , S′ , M ′ S
deltas and the orthogonality relation
ML,MS
36
L S
J
′
L S
ML MS
M ′ J
ML MS
J
MJ
= δJ ′ ,J δ M ′ J ,MJ
M ′ J
using the Kronecker
(3.118)