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Updated 2010-02-25 03:19

Here L = J + K holds.

The J and K operators listed below obey the same rules that
the L operators obey in Angular Momentum Operators.

The Ql operators listed below are defined in Singlet and Triplet Operators.
For a particular j, k pair, all the different Ql should total to the identity matrix.

## Explicit Matrices for Operators in | mJ mK > bases

| mJ mK > is a short-hand for the | j mJ > | k mK > state.

In the following matrices,
the left-most column and top-most row represent the state with mJ=+j and mK=+k,
the right-most column and bottom-most row represent the state with mJ=-j and mK=-k,
the value of mK changes between each row and column, and
the value of mJ usually stays the same between each row and column.

## Case with j=1/2 and k=1/2

Operator
Matrix in | mJ mK > basis
Jz
`    1/2     0       0       0    0       1/2     0       0    0       0      -1/2     0    0       0       0      -1/2`
Kz
`    1/2     0       0       0    0      -1/2     0       0    0       0       1/2     0    0       0       0      -1/2`
Lz
`    1       0       0       0    0       0       0       0    0       0       0       0    0       0       0      -1`
J+
`    0       0       1       0    0       0       0       1    0       0       0       0    0       0       0       0`
K+
`    0       1       0       0    0       0       0       0    0       0       0       1    0       0       0       0`
L+
`    0       1       1       0    0       0       0       1    0       0       0       1    0       0       0       0`
J-
`    0       0       0       0    0       0       0       0    1       0       0       0    0       1       0       0`
K-
`    0       0       0       0    1       0       0       0    0       0       0       0    0       0       1       0`
L-
`    0       0       0       0    1       0       0       0    1       0       0       0    0       1       1       0`

Operator
Matrix in | mJ mK > basis
Jx
`    0       0       1/2     0    0       0       0       1/2    1/2     0       0       0    0       1/2     0       0`
Kx
`    0       1/2     0       0    1/2     0       0       0    0       0       0       1/2    0       0       1/2     0`
Lx
`    0       1/2     1/2     0    1/2     0       0       1/2    1/2     0       0       1/2    0       1/2     1/2     0`
Jy
`    0       0      -i/2     0    0       0       0      -i/2    i/2     0       0       0    0       i/2     0       0`
Ky
`    0      -i/2     0       0    i/2     0       0       0    0       0       0      -i/2    0       0       i/2     0`
Ly
`    0      -i/2    -i/2     0    i/2     0       0      -i/2    i/2     0       0      -i/2    0       i/2     i/2     0`

Operator
Matrix in | mJ mK > basis
Jx Kx
`    0       0       0       1/4    0       0       1/4     0    0       1/4     0       0    1/4     0       0       0`
Jy Ky
`    0       0       0      -1/4    0       0       1/4     0    0       1/4     0       0   -1/4     0       0       0`
Jz Kz
`    1/4     0       0       0    0      -1/4     0       0    0       0      -1/4     0    0       0       0       1/4`
JK
`    1/4     0       0       0    0      -1/4     2/4     0    0       2/4    -1/4     0    0       0       0       1/4`
Lx2
`    1/2     0       0       1/2    0       1/2     1/2     0    0       1/2     1/2     0    1/2     0       0       1/2`
Ly2
`    1/2     0       0      -1/2    0       1/2     1/2     0    0       1/2     1/2     0   -1/2     0       0       1/2`
Lz2
`    1       0       0       0    0       0       0       0    0       0       0       0    0       0       0       1`
L2
`    2       0       0       0    0       1       1       0    0       1       1       0    0       0       0       2`
Q0
`    0       0       0       0    0       1/2    -1/2     0    0      -1/2     1/2     0    0       0       0       0`
Q1
`    1       0       0       0    0       1/2     1/2     0    0       1/2     1/2     0    0       0       0       1`

## Case with j=1 and k=1/2

Operator
Matrix in | mJ mK > basis
Jx
`    0       0   1/sqrt(2)   0       0       0    0       0       0   1/sqrt(2)   0       01/sqrt(2)   0       0       0   1/sqrt(2)   0    0   1/sqrt(2)   0       0       0   1/sqrt(2)    0       0   1/sqrt(2)   0       0       0    0       0       0   1/sqrt(2)   0       0`
Kx
`    0       1/2     0       0       0       0    1/2     0       0       0       0       0    0       0       0       1/2     0       0    0       0       1/2     0       0       0    0       0       0       0       0       1/2    0       0       0       0       1/2     0`
Lx
`    0       1/2 1/sqrt(2)   0       0       0    1/2     0       0   1/sqrt(2)   0       01/sqrt(2)   0       0       1/2 1/sqrt(2)   0    0   1/sqrt(2)   1/2     0       0   1/sqrt(2)    0       0   1/sqrt(2)   0       0       1/2    0       0       0   1/sqrt(2)   1/2     0`
Lx2
`    3/4     0       0   1/sqrt(2)   1/2     0    0       3/4 1/sqrt(2)   0       0       1/2    0   1/sqrt(2)   5/4     0       0   1/sqrt(2)1/sqrt(2)   0       0       5/4 1/sqrt(2)   0    1/2     0       0   1/sqrt(2)   3/4     0    0       1/2 1/sqrt(2)   0       0       3/4`
Jx Kx
`    0       0       0   1/sqrt(8)   0       0    0       0   1/sqrt(8)   0       0       0    0   1/sqrt(8)   0       0       0   1/sqrt(8)1/sqrt(8)   0       0       0   1/sqrt(8)   0    0       0       0   1/sqrt(8)   0       0    0       0   1/sqrt(8)   0       0       0`

Operator
Matrix in | mJ mK > basis
Jy
`    0       0  -i/sqrt(2)   0       0       0    0       0       0  -i/sqrt(2)   0       0i/sqrt(2)   0       0       0  -i/sqrt(2)   0    0   i/sqrt(2)   0       0       0  -i/sqrt(2)    0       0   i/sqrt(2)   0       0       0    0       0       0   i/sqrt(2)   0       0`
Ky
`    0      -i/2     0       0       0       0    i/2     0       0       0       0       0    0       0       0      -i/2     0       0    0       0       i/2     0       0       0    0       0       0       0       0      -i/2    0       0       0       0       i/2     0`
Ly
`    0      -i/2 -i/sqrt(2)  0       0       0    i/2     0       0  -i/sqrt(2)   0       0i/sqrt(2)   0       0      -i/2 -i/sqrt(2)  0    0   i/sqrt(2)   i/2     0       0  -i/sqrt(2)    0       0   i/sqrt(2)   0       0      -i/2    0       0       0   i/sqrt(2)   i/2     0`
Ly2
`    3/4     0       0  -1/sqrt(2)  -1/2     0    0       3/4 1/sqrt(2)   0       0      -1/2    0   1/sqrt(2)   5/4     0       0  -1/sqrt(2)-1/sqrt(2)  0       0       5/4 1/sqrt(2)   0   -1/2     0       0   1/sqrt(2)   3/4     0    0      -1/2 -1/sqrt(2)  0       0       3/4`
Jy Ky
`    0       0       0  -1/sqrt(8)   0       0    0       0   1/sqrt(8)   0       0       0    0   1/sqrt(8)   0       0       0  -1/sqrt(8)-1/sqrt(8)  0       0       0   1/sqrt(8)   0    0       0       0   1/sqrt(8)   0       0    0       0  -1/sqrt(8)   0       0       0`

Operator
Matrix in | mJ mK > basis
Jz
`    1       0       0       0       0       0    0       1       0       0       0       0    0       0       0       0       0       0    0       0       0       0       0       0    0       0       0       0      -1       0    0       0       0       0       0      -1`
Kz
`    1/2     0       0       0       0       0    0      -1/2     0       0       0       0    0       0       1/2     0       0       0    0       0       0      -1/2     0       0    0       0       0       0       1/2     0    0       0       0       0       0      -1/2`
Lz
`    3/2     0       0       0       0       0    0       1/2     0       0       0       0    0       0       1/2     0       0       0    0       0       0      -1/2     0       0    0       0       0       0      -1/2     0    0       0       0       0       0      -3/2`
Lz2
`    9/4     0       0       0       0       0    0       1/4     0       0       0       0    0       0       1/4     0       0       0    0       0       0       1/4     0       0    0       0       0       0       1/4     0    0       0       0       0       0       9/4`
Jz Kz
`    1/2     0       0       0       0       0    0      -1/2     0       0       0       0    0       0       0       0       0       0    0       0       0       0       0       0    0       0       0       0      -1/2     0    0       0       0       0       0       1/2`

Operator
Matrix in | mJ mK > basis
L2
`   15/4     0       0       0       0       0    0       7/4  sqrt(2)    0       0       0    0    sqrt(2)   11/4     0       0       0    0       0       0      11/4  sqrt(2)    0    0       0       0    sqrt(2)    7/4     0    0       0       0       0       0      15/4`
JK
`    1/2     0       0       0       0       0    0      -1/2 1/sqrt(2)   0       0       0    0   1/sqrt(2)   0       0       0       0    0       0       0       0   1/sqrt(2)   0    0       0       0   1/sqrt(2)  -1/2     0    0       0       0       0       0       1/2`
Q1/2
`    0       0       0       0       0       0    0       2/3 -sqrt(2/9)  0       0       0    0   -sqrt(2/9)  1/3     0       0       0    0       0       0       1/3 -sqrt(2/9)  0    0       0       0   -sqrt(2/9)  2/3     0    0       0       0       0       0       0`
Q3/2
`    1       0       0       0       0       0    0       1/3  sqrt(2/9)  0       0       0    0    sqrt(2/9)  2/3     0       0       0    0       0       0       2/3  sqrt(2/9)  0    0       0       0    sqrt(2/9)  1/3     0    0       0       0       0       0       1`