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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
J • K
    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
J • K
    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:

Adding Angular Momenta 3 and Adding Angular Momenta 4 show more cases.