Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 NineJSymbol

Source Notebook

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", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][( { {4, 4, 5}, {4, 4, 5}, {4, 4, 4} } )]
Out[1]=

Scope (3) 

NineJSymbol works with integer and half‐integer arguments:

In[2]:=
ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][( { {17/2, 19/2, 7}, {25/2, 8, 17/2}, {8, 21/2, 19/2} } )]
Out[2]=

Evaluate for large arguments:

In[3]:=
ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][( { {100, 80, 50}, {50, 100, 70}, {60, 50, 100} } )]
Out[3]=

Evaluate for inexact arguments:

In[4]:=
ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][( { {5., 0.5, 4.5}, {5., 0.5, 5.5}, {9., 1., 10.} } )]
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", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][ji] == ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][Transpose[ji]]]
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", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][ji] == ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][ji[[{3, 1, 2}]]]]
Out[6]=
In[7]:=
With[{ji = ( { {5, 1/2, 9/2}, {5, 1/2, 11/2}, {9, 1, 10} } )}, ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][ji] == ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][ji[[All, {3, 1, 2}]]]]
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", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][ji] == (-1)^ Total[ji, 2] ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][ji[[{2, 1, 3}]]]]
Out[8]=
In[9]:=
With[{ji = ( { {5, 1/2, 9/2}, {5, 1/2, 11/2}, {9, 1, 10} } )}, ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][ji] == (-1)^ Total[ji, 2] ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][ji[[All, {2, 1, 3}]]]]
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", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][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]\)\)]\)}]]
Out[10]=

Possible Issues (1) 

A message is issued and the result 0 returned for unphysical cases:

In[11]:=
ResourceFunction[ "NineJSymbol", ResourceSystemBase -> "https://www.wolframcloud.com/obj/resourcesystem/api/1.0"][( { {5, 1, 1/2}, {2, 3/2, 7}, {1, 1, 2} } )]
Out[11]=

Version History

  • 1.0.2 – 20 September 2023
  • 1.0.1 – 03 March 2021
  • 1.0.0 – 26 January 2021

Source Metadata

  • Citation:
    • Wei L., "New formula for 9-j symbols and their direct calculation." Computers in Physics, vol. 12, 632–634, 1998. DOI: 10.1063/1.168745

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