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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 and k=1

OperatorMatrix in | mJ mK > basis
Jx
`    0       0       0   1/sqrt(2)   0       0       0       0       0    0       0       0       0   1/sqrt(2)   0       0       0       0    0       0       0       0       0   1/sqrt(2)   0       0       0  1/sqrt(2)   0       0       0       0       0   1/sqrt(2)   0       0    0   1/sqrt(2)   0       0       0       0       0   1/sqrt(2)   0      0       0   1/sqrt(2)   0       0       0       0       0   1/sqrt(2)    0       0       0   1/sqrt(2)   0       0       0       0       0      0       0       0       0   1/sqrt(2)   0       0       0       0        0       0       0       0       0   1/sqrt(2)   0       0       0`
Kx
`    0   1/sqrt(2)   0       0       0       0       0       0       01/sqrt(2)   0   1/sqrt(2)   0       0       0       0       0       0    0   1/sqrt(2)   0       0       0       0       0       0       0      0       0       0       0   1/sqrt(2)   0       0       0       0    0       0       0   1/sqrt(2)   0   1/sqrt(2)   0       0       0      0       0       0       0   1/sqrt(2)   0       0       0       0        0       0       0       0       0       0       0   1/sqrt(2)   0      0       0       0       0       0       0   1/sqrt(2)   0   1/sqrt(2)    0       0       0       0       0       0       0   1/sqrt(2)   0`
Lx
`    0   1/sqrt(2)   0   1/sqrt(2)   0       0       0       0       01/sqrt(2)   0   1/sqrt(2)   0   1/sqrt(2)   0       0       0       0    0   1/sqrt(2)   0       0       0   1/sqrt(2)   0       0       0  1/sqrt(2)   0       0       0   1/sqrt(2)   0   1/sqrt(2)   0       0    0   1/sqrt(2)   0   1/sqrt(2)   0   1/sqrt(2)   0   1/sqrt(2)   0      0       0   1/sqrt(2)   0   1/sqrt(2)   0       0       0   1/sqrt(2)    0       0       0   1/sqrt(2)   0       0       0   1/sqrt(2)   0      0       0       0       0   1/sqrt(2)   0   1/sqrt(2)   0   1/sqrt(2)    0       0       0       0       0   1/sqrt(2)   0   1/sqrt(2)   0`
Lx2
`    1       0       1/2     0       1       0       1/2     0       0    0       3/2     0       1       0       1       0       1/2     0    1/2     0       1       0       1       0       0       0       1/2    0       1       0       3/2     0       1/2     0       1       0    1       0       1       0       2       0       1       0       1      0       1       0       1/2     0       3/2     0       1       0        1/2     0       0       0       1       0       1       0       1/2    0       1/2     0       1       0       1       0       3/2     0        0       0       1/2     0       1       0       1/2     0       1`
Jx Kx
`    0       0       0       0       1/2     0       0       0       0    0       0       0       1/2     0       1/2     0       0       0    0       0       0       0       1/2     0       0       0       0      0       1/2     0       0       0       0       0       1/2     0    1/2     0       1/2     0       0       0       1/2     0       1/2    0       1/2     0       0       0       0       0       1/2     0        0       0       0       0       1/2     0       0       0       0      0       0       0       1/2     0       1/2     0       0       0        0       0       0       0       1/2     0       0       0       0`

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

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

OperatorMatrix in | mJ mK > basis
L2
`    6       0       0       0       0       0       0       0       0    0       4       0       2       0       0       0       0       0    0       0       2       0       2       0       0       0       0      0       2       0       4       0       0       0       0       0    0       0       2       0       4       0       2       0       0      0       0       0       0       0       4       0       2       0        0       0       0       0       2       0       2       0       0      0       0       0       0       0       2       0       4       0        0       0       0       0       0       0       0       0       6`
JK
`    1       0       0       0       0       0       0       0       0    0       0       0       1       0       0       0       0       0    0       0      -1       0       1       0       0       0       0      0       1       0       0       0       0       0       0       0    0       0       1       0       0       0       1       0       0      0       0       0       0       0       0       0       1       0        0       0       0       0       1       0      -1       0       0      0       0       0       0       0       1       0       0       0        0       0       0       0       0       0       0       0       1`
Q0
`    0       0       0       0       0       0       0       0       0    0       0       0       0       0       0       0       0       0    0       0       1/3     0      -1/3     0       1/3     0       0      0       0       0       0       0       0       0       0       0    0       0      -1/3     0       1/3     0      -1/3     0       0      0       0       0       0       0       0       0       0       0        0       0       1/3     0      -1/3     0       1/3     0       0      0       0       0       0       0       0       0       0       0        0       0       0       0       0       0       0       0       0`
Q1
`    0       0       0       0       0       0       0       0       0    0       1/2     0      -1/2     0       0       0       0       0    0       0       1/2     0       0       0      -1/2     0       0      0      -1/2     0       1/2     0       0       0       0       0    0       0       0       0       0       0       0       0       0      0       0       0       0       0       1/2     0      -1/2     0        0       0      -1/2     0       0       0       1/2     0       0      0       0       0       0       0      -1/2     0       1/2     0        0       0       0       0       0       0       0       0       0`
Q2
`    1       0       0       0       0       0       0       0       0    0       1/2     0       1/2     0       0       0       0       0    0       0       1/6     0       1/3     0       1/6     0       0      0       1/2     0       1/2     0       0       0       0       0    0       0       1/3     0       2/3     0       1/3     0       0      0       0       0       0       0       1/2     0       1/2     0        0       0       1/6     0       1/3     0       1/6     0       0      0       0       0       0       0       1/2     0       1/2     0        0       0       0       0       0       0       0       0       1`