Updated 2009-04-04 15:53

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=3/2 and k=1/2

OperatorMatrix in | mJ mK > basis
Jx
`    0       0    sqrt(3)/2  0       0       0       0       0    0       0       0    sqrt(3)/2  0       0       0       0 sqrt(3)/2  0       0       0       1       0       0       0    0    sqrt(3)/2  0       0       0       1       0       0         0       0       1       0       0       0    sqrt(3)/2  0         0       0       0       1       0       0       0    sqrt(3)/2    0       0       0       0    sqrt(3)/2  0       0       0      0       0       0       0       0    sqrt(3)/2  0       0`
Kx
`    0       1/2     0       0       0       0       0       0    1/2     0       0       0       0       0       0       0    0       0       0       1/2     0       0       0       0    0       0       1/2     0       0       0       0       0         0       0       0       0       0       1/2     0       0         0       0       0       0       1/2     0       0       0         0       0       0       0       0       0       0       1/2    0       0       0       0       0       0       1/2     0`
Lx
`    0       1/2  sqrt(3)/2  0       0       0       0       0    1/2     0       0    sqrt(3)/2  0       0       0       0 sqrt(3)/2  0       0       1/2     1       0       0       0    0    sqrt(3)/2  1/2     0       0       1       0       0         0       0       1       0       0       1/2  sqrt(3)/2  0         0       0       0       1       1/2     0       0    sqrt(3)/2    0       0       0       0    sqrt(3)/2  0       0       1/2    0       0       0       0       0    sqrt(3)/2  1/2     0`
Lx2
`    1       0       0   sqrt(3)/2 sqrt(3)/2 0       0       0    0       1    sqrt(3)/2  0       0    sqrt(3)/2  0       0    0    sqrt(3)/2  2       0       0       1    sqrt(3)/2  0 sqrt(3)/2  0       0       2       1       0       0    sqrt(3)/2 sqrt(3)/2  0       0       1       2       0       0    sqrt(3)/2    0    sqrt(3)/2  1       0       0       2    sqrt(3)/2  0         0       0    sqrt(3)/2  0       0    sqrt(3)/2  1       0      0       0       0   sqrt(3)/2 sqrt(3)/2 0       0       1`
Jx Kx
`    0       0       0    sqrt(3)/4  0       0       0       0    0       0    sqrt(3)/4  0       0       0       0       0    0    sqrt(3)/4  0       0       0       1/2     0       0 sqrt(3)/4  0       0       0       1/2     0       0       0         0       0       0       1/2     0       0       0    sqrt(3)/4    0       0       1/2     0       0       0    sqrt(3)/4  0         0       0       0       0       0    sqrt(3)/4  0       0      0       0       0       0    sqrt(3)/4  0       0       0`

OperatorMatrix in | mJ mK > basis
Jy
`    0       0   -isqrt(3)/2 0       0       0       0       0    0       0       0  -isqrt(3)/2  0       0       0       0isqrt(3)/2  0       0       0      -i       0       0       0    0   isqrt(3)/2  0       0       0      -i       0       0         0       0       i       0       0       0  -isqrt(3)/2  0         0       0       0       i       0       0       0  -isqrt(3)/2    0       0       0       0   isqrt(3)/2  0       0       0      0       0       0       0       0   isqrt(3)/2  0       0`
Ky
`    0      -i/2     0       0       0       0       0       0    i/2     0       0       0       0       0       0       0    0       0       0      -i/2     0       0       0       0    0       0       i/2     0       0       0       0       0         0       0       0       0       0      -i/2     0       0         0       0       0       0       i/2     0       0       0         0       0       0       0       0       0       0      -i/2    0       0       0       0       0       0       i/2     0`
Ly
`    0      -i/2 -isqrt(3)/2 0       0       0       0       0    i/2     0       0  -isqrt(3)/2  0       0       0       0isqrt(3)/2  0       0      -i/2    -i       0       0       0    0   isqrt(3)/2  i/2     0       0      -i       0       0         0       0       i       0       0      -i/2 -isqrt(3)/2 0         0       0       0       i       i/2     0       0  -isqrt(3)/2    0       0       0       0   isqrt(3)/2  0       0      -i/2    0       0       0       0       0   isqrt(3)/2  i/2     0`
Ly2
`    1       0       0 -sqrt(3)/2 -sqrt(3)/2 0       0       0    0       1    sqrt(3)/2  0       0   -sqrt(3)/2  0       0    0    sqrt(3)/2  2       0       0      -1   -sqrt(3)/2  0-sqrt(3)/2  0       0       2       1       0       0   -sqrt(3)/2-sqrt(3)/2  0       0       1       2       0       0   -sqrt(3)/2    0   -sqrt(3)/2 -1       0       0       2    sqrt(3)/2  0         0       0   -sqrt(3)/2  0       0    sqrt(3)/2  1       0      0       0       0 -sqrt(3)/2 -sqrt(3)/2 0       0       1`
Jy Ky
`    0       0       0   -sqrt(3)/4  0       0       0       0    0       0    sqrt(3)/4  0       0       0       0       0    0    sqrt(3)/4  0       0       0      -1/2     0       0-sqrt(3)/4  0       0       0       1/2     0       0       0         0       0       0       1/2     0       0       0   -sqrt(3)/4    0       0      -1/2     0       0       0    sqrt(3)/4  0         0       0       0       0       0    sqrt(3)/4  0       0      0       0       0       0   -sqrt(3)/4  0       0       0`

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

OperatorMatrix in | mJ mK > basis
L2
`    6       0       0       0       0       0       0       0    0       3    sqrt(3)    0       0       0       0       0    0    sqrt(3)    5       0       0       0       0       0    0       0       0       4       2       0       0       0         0       0       0       2       4       0       0       0         0       0       0       0       0       5    sqrt(3)    0         0       0       0       0       0    sqrt(3)    3       0      0       0       0       0       0       0       0       6`
JK
`    3/4     0       0       0       0       0       0       0    0      -3/4  sqrt(3)/2  0       0       0       0       0    0    sqrt(3)/2  1/4     0       0       0       0       0    0       0       0      -1/4     1       0       0       0         0       0       0       1      -1/4     0       0       0         0       0       0       0       0       1/4  sqrt(3)/2  0         0       0       0       0       0    sqrt(3)/2 -3/4     0      0       0       0       0       0       0       0       3/4`
Q1
`    0       0       0       0       0       0       0       0    0       3/4 -sqrt(3/16) 0       0       0       0       0    0   -sqrt(3/16) 1/4     0       0       0       0       0    0       0       0       1/2    -1/2     0       0       0         0       0       0      -1/2     1/2     0       0       0         0       0       0       0       0       1/4 -sqrt(3/16) 0         0       0       0       0       0   -sqrt(3/16) 3/4     0      0       0       0       0       0       0       0       0`
Q2
`    1       0       0       0       0       0       0       0    0       1/4  sqrt(3/16) 0       0       0       0       0    0    sqrt(3/16) 3/4     0       0       0       0       0    0       0       0       1/2     1/2     0       0       0         0       0       0       1/2     1/2     0       0       0         0       0       0       0       0       3/4  sqrt(3/16) 0         0       0       0       0       0    sqrt(3/16) 1/4     0      0       0       0       0       0       0       0       1`