Wolfram Function Repository
Instant-use add-on functions for the Wolfram Language
Function Repository Resource:
Nine
Evaluate the Wigner 9-j symbol
Contributed by:
Jan Mangaldan
ResourceFunction["NineJSymbol"][{{j1,j2,j3},{j4,j5,j6},{j7,j8,j9}}] gives the values of the Wigner 9‐j symbol. |
Details
The 9‐j symbols vanish except when certain triples of the ji satisfy triangle inequalities.
The parameters of ResourceFunction["NineJSymbol"] can be integers or half‐integers.
Examples
Basic Examples (1)
Evaluate numerically:
| In[1]:= | ![ResourceFunction["NineJSymbol"][( {
{4, 4, 5},
{4, 4, 5},
{4, 4, 4}
} )]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/031c9ec9d30200db.png) |
| Out[1]= |  |
Scope (3)
NineJSymbol works with integer and half-integer arguments:
| In[2]:= | ![ResourceFunction["NineJSymbol"][( {
{17/2, 19/2, 7},
{25/2, 8, 17/2},
{8, 21/2, 19/2}
} )]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/3c9843be51b872ae.png) |
| Out[2]= |  |
Evaluate for large arguments:
| In[3]:= | ![ResourceFunction["NineJSymbol"][( {
{100, 80, 50},
{50, 100, 70},
{60, 50, 100}
} )]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/6eec407ce154335c.png) |
| Out[3]= |  |
Evaluate for inexact arguments:
| In[4]:= | ![ResourceFunction["NineJSymbol"][( {
{5., 0.5, 4.5},
{5., 0.5, 5.5},
{9., 1., 10.}
} )]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/0eddedb9b93c559b.png) |
| Out[4]= |  |
Properties and Relations (3)
NineJSymbol is invariant under transposition:
| In[5]:= | ![With[{ji = ( {
{5, 1/2, 9/2},
{5, 1/2, 11/2},
{9, 1, 10}
} )}, ResourceFunction["NineJSymbol"][ji] == ResourceFunction["NineJSymbol"][Transpose[ji]]]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/32d3e4af2b40bde1.png) |
| Out[5]= |  |
NineJSymbol is invariant under an even permutation of its rows or columns:
| In[6]:= | ![With[{ji = ( {
{5, 1/2, 9/2},
{5, 1/2, 11/2},
{9, 1, 10}
} )}, ResourceFunction["NineJSymbol"][ji] == ResourceFunction["NineJSymbol"][ji[[{3, 1, 2}]]]]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/3871b38c4ea66394.png) |
| Out[6]= |  |
| In[7]:= | ![With[{ji = ( {
{5, 1/2, 9/2},
{5, 1/2, 11/2},
{9, 1, 10}
} )}, ResourceFunction["NineJSymbol"][ji] == ResourceFunction["NineJSymbol"][ji[[All, {3, 1, 2}]]]]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/7a2e98a44fc60642.png) |
| Out[7]= |  |
Under an odd permutation of its rows or columns, NineJSymbol gains an extra phase factor:
| In[8]:= | ![With[{ji = ( {
{5, 1/2, 9/2},
{5, 1/2, 11/2},
{9, 1, 10}
} )}, ResourceFunction["NineJSymbol"][ji] == (-1)^
Total[ji, 2] ResourceFunction["NineJSymbol"][ji[[{2, 1, 3}]]]]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/1abd6379b07637c9.png) |
| Out[8]= |  |
| In[9]:= | ![With[{ji = ( {
{5, 1/2, 9/2},
{5, 1/2, 11/2},
{9, 1, 10}
} )}, ResourceFunction["NineJSymbol"][ji] == (-1)^
Total[ji, 2] ResourceFunction["NineJSymbol"][ji[[All, {2, 1, 3}]]]]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/7d4e4a5c3ad312b7.png) |
| Out[9]= |  |
When one of the entries is 0, NineJSymbol can be expressed in terms of SixJSymbol:
| In[10]:= | ![With[{ji = ( {
{1, 2, 3},
{1, 2, 3},
{2, 2, 0}
} )},
ResourceFunction["NineJSymbol"][ji] == (-1)^(
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(2\[InvisibleComma]1\)\(\[RightDoubleBracket]\)\)]\) +
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(1\[InvisibleComma]2\)\(\[RightDoubleBracket]\)\)]\) +
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(3\[InvisibleComma]1\)\(\[RightDoubleBracket]\)\)]\) +
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(1\[InvisibleComma]3\)\(\[RightDoubleBracket]\)\)]\))/Sqrt[(2
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(1\[InvisibleComma]3\)\(\[RightDoubleBracket]\)\)]\) + 1) (2
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(3\[InvisibleComma]1\)\(\[RightDoubleBracket]\)\)]\) + 1)]
KroneckerDelta[
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(1\[InvisibleComma]3\)\(\[RightDoubleBracket]\)\)]\),
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(2\[InvisibleComma]3\)\(\[RightDoubleBracket]\)\)]\)] KroneckerDelta[
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(3\[InvisibleComma]1\)\(\[RightDoubleBracket]\)\)]\),
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(3\[InvisibleComma]2\)\(\[RightDoubleBracket]\)\)]\)] SixJSymbol[
First[ji], {
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(2\[InvisibleComma]2\)\(\[RightDoubleBracket]\)\)]\),
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(2\[InvisibleComma]1\)\(\[RightDoubleBracket]\)\)]\),
\!\(\*SubscriptBox[\(ji\), \(\(\[LeftDoubleBracket]\)\(3\[InvisibleComma]1\)\(\[RightDoubleBracket]\)\)]\)}]]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/638cc54b95e64062.png) |
| Out[10]= |  |
Possible Issues (1)
A message is issued and the result 0 returned for unphysical cases:
| In[11]:= | ![ResourceFunction["NineJSymbol"][( {
{5, 1, 1/2},
{2, 3/2, 7},
{1, 1, 2}
} )]](https://www.wolframcloud.com/obj/resourcesystem/images/d7f/d7f82fc3-b09e-4d92-882b-2a05caed294b/1-0-1/6057b0fbfb08c200.png) |
| Out[11]= |  |
Related Links
Version History
- 1.0.2 – 20 September 2023
- 1.0.1 – 03 March 2021
- 1.0.0 – 26 January 2021
Source Metadata
Related Resources
Related Symbols
License Information
This work is licensed under a Creative Commons Attribution 4.0 International License