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Adding Angular Momenta 4

rename
Updated 2009-04-04 15:49

Adding Angular Momenta

Please study Adding Angular Momenta before reading the page below.



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       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 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       0
1/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
memorize

 

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       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 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       0
i/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
memorize

 

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
memorize

 

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
memorize

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