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Adding Angular Momenta 1

rename
Updated 2009-03-23 14:50

Before you proceed

Please study Adding Angular Momenta before reading the page below.

Case with j=1, k=1, and l=0, 1, or 2.

l=0 gives 1 singlet state with mL=0.
l=1 gives 3 triplet states with mL=-1, 0, 1.
l=2 gives 5 quintet states with mL=-2, -1, 0, 1, 2.
(2 j + 1)(2 k + 1) gives 9 states in all.

 

| l mL >
| mJ mK > = | j mJ > | k mK >
| 2  2 >                    |  1  1 >
| 2  1 >      sqrt(3/6) |  1  0 > + sqrt(3/6) |  0  1 >
| 2  0 >      sqrt(1/6) |  1 -1 > + sqrt(1/6) | -1  1 >
   + sqrt(4/6) |  0  0 >
| 2 -1 >      sqrt(3/6) | -1  0 > + sqrt(3/6) |  0 -1 >
| 2 -2 >                    | -1 -1 >
| 1  1 >      sqrt(3/6) |  1  0 > - sqrt(3/6) |  0  1 >
| 1  0 >      sqrt(3/6) |  1 -1 > - sqrt(3/6) | -1  1 >
| 1 -1 >      sqrt(3/6) | -1  0 > - sqrt(3/6) |  0 -1 >
| 0  0 >      sqrt(2/6) |  1 -1 > + sqrt(2/6) | -1  1 >
    - sqrt(2/6) |  0  0 >
memorize

Case with j=3/2, k=1, and l=1/2, 3/2, or 5/2.

l=1/2 gives 2 doublet states with mL=-1/2. 1/2.
l=3/2 gives 4 quadruplet states with mL=-3/2, -1/2, 1/2, 3/2.
l=5/2 gives 6 sextuplet states with mL=-5/2, -3/2, -1/2, 1/2, 3/2, 5/2.
(2 j + 1)(2 k + 1) gives 12 states in all.

 

| l mL >
| mJ mK > = | j mJ > | k mK >
| 5/2  5/2 >                    |  3/2  1 >
| 5/2  3/2 >   sqrt(12/30) |  3/2  0 > + sqrt(18/30) |  1/2  1 >
| 5/2  1/2 >     sqrt(3/30) |  3/2 -1 > + sqrt(18/30) |  1/2  0 >
  + sqrt(9/30) | -1/2  1 >
| 5/2 -1/2 >     sqrt(3/30) | -3/2  1 > + sqrt(18/30) | -1/2  0 >
  + sqrt(9/30) |  1/2 -1 >
| 5/2 -3/2 >   sqrt(12/30) | -3/2  0 > + sqrt(18/30) | -1/2 -1 >
| 5/2 -5/2 >                    | -3/2 -1 >
| 3/2  3/2 >   sqrt(18/30) |  3/2  0 >   - sqrt(12/30) |  1/2  1 >
| 3/2  1/2 >   sqrt(12/30) |  3/2 -1 >   + sqrt(2/30) |  1/2  0 >
- sqrt(16/30) | -1/2  1 >
| 3/2 -1/2 >   sqrt(12/30) | -3/2  1 >   + sqrt(2/30) | -1/2  0 >
- sqrt(16/30) |  1/2 -1 >
| 3/2 -3/2 >   sqrt(18/30) | -3/2  0 >   - sqrt(12/30) | -1/2 -1 >
| 1/2  1/2 >   sqrt(15/30) |  3/2 -1 >   - sqrt(10/30) |  1/2  0 >
  + sqrt(5/30) | -1/2  1 >
| 1/2 -1/2 >   sqrt(15/30) | -3/2  1 >   - sqrt(10/30) | -1/2  0 >
  + sqrt(5/30) |  1/2 -1 >
memorize

Case with j=2, k=1, and l=1, 2, or 3.

l=1 gives 3 triplet states with mL=-1, 0, 1.
l=2 gives 5 quintet states with mL=-2, -1, 0, 1, 2.
l=3 gives 7 septuplet states with mL=-3, -2, -1, 0, 1, 2, 3.
(2 j + 1)(2 k + 1) gives 15 states in all.

 

| l mL >
| mJ mK > = | j mJ > | k mK >
| 3 3 >                    |  2  1 >
| 3  2 >   sqrt(10/30) |  2  0 > + sqrt(20/30) |  1  1 >
| 3  1 >   sqrt(16/30) |  1  0 > + sqrt(12/30) |  0  1 >
  + sqrt(2/30) |  2 -1 >
| 3  0 >     sqrt(6/30) |  1 -1 >   + sqrt(6/30) | -1  1 >
+ sqrt(18/30) |  0  0 >
| 3 -1 >     sqrt(2/30) | -2  1 > + sqrt(16/30) | -1 0 >
+ sqrt(12/30) |  0   -1 >
| 3 -2 >   sqrt(10/30) | -2  0 > + sqrt(20/30) | -1 -1 >
| 3 -3 >                    | -2 -1 >
| 2  2 >   sqrt(20/30) |  2  0 >   - sqrt(10/30) |  1  1 >
| 2  1 >     sqrt(5/30) |  1  0 >   - sqrt(15/30) |  0  1 >
+ sqrt(10/30) |  2 -1 >
| 2  0 >   sqrt(15/30) |  1 -1 >   - sqrt(15/30) | -1  1 >
| 2 -1 >     sqrt(5/30) | -1  0 >   - sqrt(15/30) |  0 -1 >
+ sqrt(10/30) | -2  1 >
| 2 -2 >   sqrt(20/30) | -2  0 >   - sqrt(10/30) | -1 -1 >
| 1  1 >   - sqrt(9/30) |  1  0 >   + sqrt(3/30) |  0  1 >
+ sqrt(18/30) |  2 -1 >
| 1  0 >     sqrt(9/30) |  1 -1 >   + sqrt(9/30) | -1  1 >
  - sqrt(12/30) |  0  0 >
| 1 -1 >   - sqrt(9/30) | -1  0 >   + sqrt(3/30) |  0 -1 >
+ sqrt(18/30) | -2  1 >
memorize

Case with j=3/2, k=3/2, and l=0, 1, 2, 3.

l=0 gives 1 singlet states with mL=0.
l=1 gives 3 triplet states with mL=-1, 0, 1.
l=2 gives 5 quintet states with mL=-2, -1, 0, 1, 2.
l=3 gives 7 septuplet states with mL=-3, -2, -1, 0, 1, 2, 3.
(2 j + 1)(2 k + 1) gives 16 states in all.

 

| l mL >
| mJ mK > = | j mJ > | k mK >
| 3  3 >                    |  3/2  3/2 >
| 3  2 >   sqrt(10/20) |  3/2  1/2 > + sqrt(10/20) |  1/2  3/2 >
| 3  1 >    sqrt(4/20) |  3/2 -1/2 >   + sqrt(4/20) | -1/2  3/2 >
+ sqrt(12/20) |  1/2  1/2 >
| 3  0 >    sqrt(1/20) |  3/2 -3/2 >   + sqrt(9/20) |  1/2 -1/2 >
+ sqrt(1/20) | -3/2  3/2 >   + sqrt(9/20) | -1/2  1/2 >
| 3 -1 >    sqrt(4/20) | -3/2  1/2 >   + sqrt(4/20) |  1/2 -3/2 >
+ sqrt(12/20) | -1/2 -1/2 >
| 3 -2 >   sqrt(10/20) | -3/2 -1/2 > + sqrt(10/20) | -1/2 -3/2 >
| 3 -3 >                    | -3/2 -3/2 >
| 2  2 >   sqrt(10/20) |  3/2  1/2 > - sqrt(10/20) |  1/2  3/2 >
| 2  1 >   sqrt(10/20) |  3/2 -1/2 > - sqrt(10/20) | -1/2  3/2 >
| 2  0 >    sqrt(5/20) |  3/2 -3/2 >   + sqrt(5/20) |  1/2 -1/2 >
  - sqrt(5/20) | -3/2  3/2 >   - sqrt(5/20) | -1/2  1/2 >
| 2 -1 >   sqrt(10/20) | -3/2  1/2 > - sqrt(10/20) |  1/2 -3/2 >
| 2 -2 >   sqrt(10/20) | -3/2 -1/2 > - sqrt(10/20) | -1/2 -3/2 >
| 1  1 >    sqrt(6/20) |  3/2 -1/2 >   + sqrt(6/20) | -1/2  3/2 >
  - sqrt(8/20) |  1/2  1/2 >
| 1  0 >    sqrt(9/20) |  3/2 -3/2 >   - sqrt(1/20) |  1/2 -1/2 >
+ sqrt(9/20) | -3/2  3/2 >   - sqrt(1/20) | -1/2  1/2 >
| 1 -1 >    sqrt(6/20) | -3/2  1/2 >   + sqrt(6/20) |  1/2 -3/2 >
  - sqrt(8/20) | -1/2 -1/2 >
| 0  0 >    sqrt(5/20) |  3/2 -3/2 >   - sqrt(5/20) |  1/2 -1/2 >
  - sqrt(5/20) | -3/2  3/2 >   + sqrt(5/20) | -1/2  1/2 >
memorize

References:

http://en.wikipedia.org/wiki/Table_of_Clebsch-Gordan_coefficients
gives cases for larger j, k values.

See Also:

For cases with k=1/2, see Adding Angular Momenta.
When you feel comfortable with the pages above,
see Adding Angular Momenta 2 and Singlet and Triplet Operators.