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Adding Angular Momenta 2

rename
Updated 2010-02-25 03:19

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=1/2 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
memorize

 

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
memorize

 

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
memorize

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 0
1/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 0
1/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
memorize

 

Operator
Matrix in | mJ mK > basis
Jy
    0       0  -i/sqrt(2)   0       0       0
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 -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 0
i/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
memorize

 

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
memorize

 

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
memorize

See Also: