9j symbols according to Eugene Wigner are used to couple four angular impulses in quantum mechanics .
Accordingly, the 9j symbol is defined as follows via the coupling coefficient:
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{\ displaystyle {\ sqrt {(2j_ {3} +1) (2j_ {6} +1) (2j_ {7} +1) (2j_ {8} +1)}} {\ begin {Bmatrix} j_ {1 } & j_ {2} & j_ {3} \\ j_ {4} & j_ {5} & j_ {6} \\ j_ {7} & j_ {8} & j_ {9} \ end {B matrix}} = \ langle ((j_ { 1} j_ {2}) j_ {3}, (j_ {4} j_ {5}) j_ {6}) j_ {9} | ((j_ {1} j_ {4}) j_ {7}, (j_ {2} j_ {5}) j_ {8}) j_ {9} \ rangle.}
The Umkopplungskoeffizient on the right side transformed between two base sets: in a will to be coupled with to and then and to . In the other is to be coupled with to and then and to .
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{\ displaystyle | ((j_ {1} j_ {4}) j_ {7}, (j_ {2} j_ {5}) j_ {8}) j_ {9} \ rangle = \ sum _ {j_ {3} } \ sum _ {j_ {6}} \ langle ((j_ {1} j_ {2}) j_ {3}, (j_ {4} j_ {5}) j_ {6}) j_ {9} | (( j_ {1} j_ {4}) j_ {7}, (j_ {2} j_ {5}) j_ {8}) j_ {9} \ rangle \, | ((j_ {1} j_ {2}) j_ {3}, (j_ {4} j_ {5}) j_ {6}) j_ {9} \ rangle}
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{\ displaystyle = {\ sqrt {(2j_ {7} +1) (2j_ {8} +1)}} \, \ sum _ {j_ {3}} \ sum _ {j_ {6}} \, {\ sqrt {(2j_ {3} +1) (2j_ {6} +1)}} {\ begin {Bmatrix} j_ {1} & j_ {2} & j_ {3} \\ j_ {4} & j_ {5} & j_ { 6} \\ j_ {7} & j_ {8} & j_ {9} \ end {Bmatrix}} | ((j_ {1} j_ {2}) j_ {3}, (j_ {4} j_ {5}) j_ {6}) j_ {9} \ rangle}
Symmetries
The 9j symbol is invariant under reflection on its diagonals and with even permutation of the rows or columns:
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{\ displaystyle {\ begin {Bmatrix} j_ {1} & j_ {2} & j_ {3} \\ j_ {4} & j_ {5} & j_ {6} \\ j_ {7} & j_ {8} & j_ {9} \ end {Bmatrix}} = {\ begin {Bmatrix} j_ {1} & j_ {4} & j_ {7} \\ j_ {2} & j_ {5} & j_ {8} \\ j_ {3} & j_ {6} & j_ { 9} \ end {Bmatrix}} = {\ begin {Bmatrix} j_ {9} & j_ {6} & j_ {3} \\ j_ {8} & j_ {5} & j_ {2} \\ j_ {7} & j_ {4 } & j_ {1} \ end {Bmatrix}} = {\ begin {Bmatrix} j_ {7} & j_ {4} & j_ {1} \\ j_ {9} & j_ {6} & j_ {3} \\ j_ {8} & j_ {5} & j_ {2} \ end {Bmatrix}}.}
In the case of odd permutations of rows or columns, the phase factor is multiplied by . Example:
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{\ displaystyle {\ begin {Bmatrix} j_ {1} & j_ {2} & j_ {3} \\ j_ {4} & j_ {5} & j_ {6} \\ j_ {7} & j_ {8} & j_ {9} \ end {Bmatrix}} = (- 1) ^ {S} {\ begin {Bmatrix} j_ {4} & j_ {5} & j_ {6} \\ j_ {1} & j_ {2} & j_ {3} \\ j_ { 7} & j_ {8} & j_ {9} \ end {Bmatrix}} = (- 1) ^ {S} {\ begin {Bmatrix} j_ {2} & j_ {1} & j_ {3} \\ j_ {5} & j_ {4} & j_ {6} \\ j_ {8} & j_ {7} & j_ {9} \ end {Bmatrix}}.}
Attributed to 6j symbols
The 9j symbols can be expressed as sums over products of three 6j symbols :
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{\ displaystyle {\ begin {Bmatrix} j_ {1} & j_ {2} & j_ {3} \\ j_ {4} & j_ {5} & j_ {6} \\ j_ {7} & j_ {8} & j_ {9} \ end {Bmatrix}} = \ sum _ {x} (- 1) ^ {2x} (2x + 1) {\ begin {Bmatrix} j_ {1} & j_ {4} & j_ {7} \\ j_ {8} & j_ {9} & x \ end {Bmatrix}} {\ begin {Bmatrix} j_ {2} & j_ {5} & j_ {8} \\ j_ {4} & x & j_ {6} \ end {Bmatrix}} {\ begin {Bmatrix} j_ {3} & j_ {6} & j_ {9} \\ x & j_ {1} & j_ {2} \ end {Bmatrix}}}
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In doing so, a total is made of all those for which the triangle condition is fulfilled for the factors (see 3j symbol or 6j symbol ).
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Special case
A special case is if the 9j symbol is proportional to a 6j symbol:
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{\ displaystyle {\ begin {Bmatrix} j_ {1} & j_ {2} & j_ {3} \\ j_ {4} & j_ {5} & j_ {6} \\ j_ {7} & j_ {8} & 0 \ end {Bmatrix }} = {\ frac {\ delta _ {j_ {3}, j_ {6}} \ delta _ {j_ {7}, j_ {8}}} {\ sqrt {(2j_ {3} +1) (2j_ {7} +1)}}} (- 1) ^ {j_ {2} + j_ {3} + j_ {4} + j_ {7}} {\ begin {Bmatrix} j_ {1} & j_ {2} & j_ {3} \\ j_ {5} & j_ {4} & j_ {7} \ end {Bmatrix}}.}
Orthogonality relation
The 9j symbols satisfy the orthogonality relation:
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{\ displaystyle \ sum _ {j_ {7} j_ {8}} (2j_ {7} +1) (2j_ {8} +1) {\ begin {Bmatrix} j_ {1} & j_ {2} & j_ {3} \\ j_ {4} & j_ {5} & j_ {6} \\ j_ {7} & j_ {8} & j_ {9} \ end {Bmatrix}} {\ begin {Bmatrix} j_ {1} & j_ {2} & j_ { 3} '\\ j_ {4} & j_ {5} & j_ {6}' \\ j_ {7} & j_ {8} & j_ {9} \ end {Bmatrix}} = {\ frac {\ delta _ {j_ {3 } j_ {3} '} \ delta _ {j_ {6} j_ {6}'} {\ begin {Bmatrix} j_ {1} & j_ {2} & j_ {3} \ end {Bmatrix}} {\ begin {Bmatrix } j_ {4} & j_ {5} & j_ {6} \ end {Bmatrix}} {\ begin {Bmatrix} j_ {3} & j_ {6} & j_ {9} \ end {Bmatrix}}} {(2j_ {3} +1) (2j_ {6} +1)}}.}
The triangular delta is defined as with the 3j symbol and expresses compliance with the triangle condition.
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{\ displaystyle {\ begin {Bmatrix} j_ {1} & j_ {2} & j_ {3} \ end {Bmatrix}}}
literature
Alan Robert Edmonds : Angular momentum in quantum mechanics, BI university pocket books 1964 (English original Princeton UP 1957)
A. Messiah : Quantum Mechanics , Volume 2, De Gruyter 1985, Appendix C.
Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">