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## Adding Angular Momenta in Quantum Mechanics

Here we will add the angular momenta vectors
J and K to get L = J + K.

Each of J, K, L behaves like
L = Lx x + Ly y + Lz z

where Lx, Ly, Lz
are operators as in Angular Momentum Operators

and x, y, z are unit vectors
along the cartesian coordinate axes x, y, z.

Each of J, K can represent:
an electron spin (often denoted S with S2=s(s+1)=3/4, s=1/2, Sz=mS=-1/2 or +1/2),
a nuclear spin (often denoted I),
an angular momentum (often denoted L), or
any combination of the above.

In the following, the left-hand column equals the right-hand column.

Left ColumnRight Column
LxJx + Kx
LyJy + Ky
LzJz + Kz
L+ = Lx + i LyJ+ + K+
L- = Lx - i LyJ- + K-
Jz | j mJ >mJ | j mJ >
Kz | k mK >mK | k mK >
Lz | l mL >mL | l mL >
mJ-j, -j+1, -j+2, ..., j-2, j-1, j
mK-k, -k+1, -k+2, ..., k-2, k-1, k
mL-l, -l+1, -l+2, ..., l-2, l-1, l
mLmJ + mK
mL-j-k, -j-k+1, -j-k+2, ..., j+k-2, j+k-1, j+k
2 j + 1Number of different | j mJ > states for a particular j
2 k + 1Number of different | k mK > states for a particular k
2 l + 1Number of different | l mL > states for a particular l
(2 j + 1)(2 k + 1)Number of different | mJ mK > = | j mJ > | k mK >
states for a particular j and k

Left ColumnRight Column
l |j-k|, |j-k|+1, |j-k|+2, ..., j+k-2, j+k-1, j+k
|j-k|Minimum value of l
j+kMaximum value of l
J2 | j mJ > = JJ | j mJ >j(j+1) | j mJ >
K2 | k mK > = KK | k mK >k(k+1) | k mK >
L2 | l mL > = LL | l mL >l(l+1) | l mL >
LLLx2 + Ly2 + Lz2
LL(J + K)2
LLJJ + 2 JK + KK
2 JKL2 - J2 - K2
2 JKl(l+1) - j(j+1) - k(k+1)
JK Jx Kx + Jy Ky + Jz Kz
Jx Kx + Jy KyJK - Jz Kz
Jx Kx + Jy Ky(1/2)[l(l+1) - j(j+1) - k(k+1)] - mJ mK
< j' m'J | j mJ >1 if j=j' and mJ=m'J, 0 otherwise
< k' m'K | k mK >1 if k=k' and mK=m'K, 0 otherwise
< l' m'L | l mL >1 if l=l' and mL=m'L, 0 otherwise
[A, B]-A B - B A, the commutator of the matrices A and B
[Jv, Kw]-0 for v,w=x,y,z,+,-
[Jv, Lw]-[Jv, Jw]- for v,w=x,y,z,+,-
[Kv, Lw]-[Kv, Kw]- for v,w=x,y,z,+,-

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

l=0 gives 1 singlet state with mL=0.
l=1 gives 3 triplet states with mL=-1, 0, 1.
(2 j + 1)(2 k + 1) gives 4 states in all.

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

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

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.
(2 j + 1)(2 k + 1) gives 6 states in all.

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

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

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 8 states in all.

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

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

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 10 states in all.

| l mL >
| mJ mK > = | j mJ > | k mK >
| 5/2  5/2 >                    |  2  1/2 >
| 5/2  3/2 >      sqrt(1/5) |  2 -1/2 > + sqrt(4/5) |  1  1/2 >
| 5/2  1/2 >      sqrt(2/5) |  1 -1/2 > + sqrt(3/5) |  0  1/2 >
| 5/2 -1/2 >      sqrt(2/5) | -1  1/2 > + sqrt(3/5) |  0 -1/2 >
| 5/2 -3/2 >      sqrt(1/5) | -2  1/2 > + sqrt(4/5) | -1 -1/2 >
| 5/2 -5/2 >                    | -2 -1/2 >
| 3/2  3/2 >      sqrt(4/5) |  2 -1/2 > - sqrt(1/5) |  1  1/2 >
| 3/2  1/2 >      sqrt(3/5) |  1 -1/2 > - sqrt(2/5) |  0  1/2 >
| 3/2 -1/2 >      sqrt(3/5) | -1  1/2 > - sqrt(2/5) |  0 -1/2 >
| 3/2 -3/2 >      sqrt(4/5) | -2  1/2 > - sqrt(1/5) | -1 -1/2 >

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

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 12 states in all.

| l mL >
| mJ mK > = | j mJ > | k mK >
| 3  3 >                    |  5/2  1/2 >
| 3  2 >     sqrt(5/30) |  5/2 -1/2 > + sqrt(25/30) |  3/2  1/2 >
| 3  1 >   sqrt(10/30) |  3/2 -1/2 > + sqrt(20/30) |  1/2  1/2 >
| 3  0 >   sqrt(15/30) |  1/2 -1/2 > + sqrt(15/30) | -1/2  1/2 >
| 3 -1 >   sqrt(10/30) | -3/2  1/2 > + sqrt(20/30) | -1/2 -1/2 >
| 3 -2 >     sqrt(5/30) | -5/2  1/2 > + sqrt(25/30) | -3/2 -1/2 >
| 3 -3 >                    | -5/2 -1/2 >
| 2  2 >   sqrt(25/30) |  5/2 -1/2 >   - sqrt(5/30) |  3/2  1/2 >
| 2  1 >   sqrt(20/30) |  3/2 -1/2 > - sqrt(10/30) |  1/2  1/2 >
| 2  0 >   sqrt(15/30) |  1/2 -1/2 > - sqrt(15/30) | -1/2  1/2 >
| 2 -1 >   sqrt(20/30) | -3/2  1/2 > - sqrt(10/30) | -1/2 -1/2 >
| 2 -2 >   sqrt(25/30) | -5/2  1/2 >   - sqrt(5/30) | -3/2 -1/2 >

## References:

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