Adding Angular Momenta
rename
Updated
2009-03-23 14:50
Adding Angular Momenta in Quantum Mechanics
Here we will add the angular momenta vectorsJ 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 Column | Right Column |
|---|---|
| Lx | Jx + Kx |
| Ly | Jy + Ky |
| Lz | Jz + Kz |
| L+ = Lx + i Ly | J+ + K+ |
| L- = Lx - i Ly | J- + 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 |
| mL | mJ + mK |
| mL | -j-k, -j-k+1, -j-k+2, ..., j+k-2, j+k-1, j+k |
| 2 j + 1 | Number of different | j mJ > states for a particular j |
| 2 k + 1 | Number of different | k mK > states for a particular k |
| 2 l + 1 | Number 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 Column | Right 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+k | Maximum value of l |
| J2 | j mJ > = J • J | j mJ > | j(j+1) | j mJ > |
| K2 | k mK > = K • K | k mK > | k(k+1) | k mK > |
| L2 | l mL > = L • L | l mL > | l(l+1) | l mL > |
| L • L | Lx2 + Ly2 + Lz2 |
| L • L | (J + K)2 |
| L • L | J • J + 2 J • K + K • K |
| 2 J • K | L2 - J2 - K2 |
| 2 J • K | l(l+1) - j(j+1) - k(k+1) |
| J • K | Jx Kx + Jy Ky + Jz Kz |
| Jx Kx + Jy Ky | J • K - 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.
gives cases for larger j, k values.
See Also:
For cases with k=1 or 3/2, see Adding Angular Momenta 1.
When you feel comfortable with the pages above,
see Adding Angular Momenta 2 and Singlet and Triplet Operators.
see Adding Angular Momenta 2 and Singlet and Triplet Operators.
