GellMann

GellMann
Produces a Gell-Mann operator
Other toolboxes required	none
Related functions	GenGellMann GenPauli Pauli
Function category	Special states, vectors, and operators

GellMann is a function that produces the 3-by-3 Gell-Mann matrices, as defined here:

\(g_0 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)	\(g_1 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)	\(g_2 = \begin{bmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)	\(g_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)	\(g_4 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}\)
	\(g_5 = \begin{bmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{bmatrix}\)	\(g_6 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}\)	\(g_7 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{bmatrix}\)	\(g_8 = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{bmatrix}\)

Syntax

• G = GellMann(IND)
• G = GellMann(IND,SP)

Argument descriptions

• IND: An index indicating which Gell-Mann operator you would like to be generated. Should be an integer between 0 and 8, inclusive.
• SP (optional, default 0): A flag (either 1 or 0) indicating that the Gell-Mann operator produced should or should not be sparse.

Examples

>> GellMann(1)
 
ans =
 
     0     1     0
     1     0     0
     0     0     0
 
>> GellMann(3,1) % the matrix produced here will be sparse
 
ans =
 
   (1,1)        1
   (2,2)       -1

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  GELLMANN    Produces a Gell-Mann operator

2. %   This function has one required argument:

3. %     IND (an integer between 0 and 8, inclusive)

4. %

5. %   G = GellMann(IND) is the 3-by-3 Gell-Mann matrix indicated by the value

6. %   of IND. IND = 0 gives the identity matrix, while values of 1 through 8

7. %   each indicate one of the other 8 Gell-Mann matrices.

8. %

9. %   This function has one optional argument:

10. %     SP (default 0)

11. %

12. %   G = GellMann(IND,SP) is as above, with sparsity of the output

13. %   determined by the value of SP. If SP = 0 then the output will be full,

14. %   if SP = 1 then the output will be sparse.

15. %

16. %   URL: http://www.qetlab.com/GellMann

18. %   requires: opt_args.m

19. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)

20. %   package: QETLAB

21. %   last updated: December 18, 2013

23. function g = GellMann(ind,varargin)

25. % set optional argument defaults: sp=0

26. [sp] = opt_args({ 0 },varargin{:});

28. if(ind == 1)

29.     g = [0 1 0;1 0 0;0 0 0];

30. elseif(ind == 2)

31.     g = [0 -1i 0;1i 0 0;0 0 0];

32. elseif(ind == 3)

33.     g = [1 0 0;0 -1 0;0 0 0];

34. elseif(ind == 4)

35.     g = [0 0 1;0 0 0;1 0 0];

36. elseif(ind == 5)

37.     g = [0 0 -1i;0 0 0;1i 0 0];

38. elseif(ind == 6)

39.     g = [0 0 0;0 0 1;0 1 0];

40. elseif(ind == 7)

41.     g = [0 0 0;0 0 -1i;0 1i 0];

42. elseif(ind == 8)

43.     g = [1 0 0;0 1 0;0 0 -2]/sqrt(3);

44. else

45.     g = eye(3);

46. end

48. if(sp)

49.     g = sparse(g);

50. end

External links

• Gell-Mann matrices at Wikipedia