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Adding Angular Momenta 3

rename
Updated 2009-04-04 15:53

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=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
memorize

 

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 0
isqrt(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 0
isqrt(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
memorize

 

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
memorize

 

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
memorize

See Also: