Table of Clebsch–Gordan coefficients
This is a table of Clebsch–Gordan coefficients used for adding angular momentum values in quantum mechanics. The overall sign of the coefficients for each set of constant
,
,
is arbitrary to some degree and has been fixed according to the Condon-Shortley and Wigner sign convention as discussed by Baird and Biedenharn.[1] Tables with the same sign convention may be found in the Particle Data Group's Review of Particle Properties[2] and in online tables.[3]
Formulation
The Clebsch–Gordan coefficients are the solutions to
![|(j_1j_2)jm\rangle = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2}
|j_1m_1j_2m_2\rangle \langle j_1j_2;m_1m_2|j_1j_2;jm\rangle](../I/m/befa953e6b50226e6501dfb1202cef0349c2a6ec.svg)
Explicitly:
![\begin{align}
\langle j_1j_2;m_1m_2|j_1j_2;jm\rangle = \
&\delta_{m,m_1+m_2} \sqrt{\frac{(2j+1)(j+j_1-j_2)!(j-j_1+j_2)!(j_1+j_2-j)!}{(j_1+j_2+j+1)!}}\ \times \\
&\sqrt{(j+m)!(j-m)!(j_1-m_1)!(j_1+m_1)!(j_2-m_2)!(j_2+m_2)!}\ \times \\
&\sum_k \frac{(-1)^k}{k!(j_1+j_2-j-k)!(j_1-m_1-k)!(j_2+m_2-k)!(j-j_2+m_1+k)!(j-j_1-m_2+k)!}.
\end{align}](../I/m/83c6c820e2f934f0de376cbbe1f90ce0ebbd1fde.svg)
The summation is extended over all integer k for which the argument of every factorial is nonnegative.[4]
For brevity, solutions with m < 0 and j1 < j2 are omitted. They may be calculated using the simple relations
.
and
.
A complete list [5]
j2=0
When j2 = 0, the Clebsch–Gordan coefficients are given by
.
j1=1/2, j2=1/2
m=1 |
j |
m1, m2 |
|
1 |
1/2, 1/2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
m=-1 |
j |
m1, m2 |
|
1 |
-1/2, -1/2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
j1=1, j2=1/2
m=3/2 |
j |
m1, m2 |
|
3/2 |
1, 1/2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
j1=1, j2=1
m=2 |
j |
m1, m2 |
|
2 |
1, 1 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
j1=3/2, j2=1/2
m=2 |
j |
m1, m2 |
|
2 |
3/2, 1/2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
j1=3/2, j2=1
m=5/2 |
j |
m1, m2 |
|
5/2 |
3/2, 1 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
m=1/2 |
j |
m1, m2 |
|
5/2 |
3/2 |
1/2 |
3/2, -1 |
![\sqrt{\frac{1}{10}}\!\,](../I/m/e461f3044f6531ac49afc67501bcc70e1ef20051.svg) |
|
![\sqrt{\frac{1}{2}}\!\,](../I/m/af39f634eb891de4573e5b2e486b13a6d2df7ea4.svg) |
1/2, 0 |
![\sqrt{\frac{3}{5}}\!\,](../I/m/a3c6237454cec1941476de55a9b34e4b3cbbd843.svg) |
![\sqrt{\frac{1}{15}}\!\,](../I/m/dee88493dbf645c6f4079065117665d8f93ba578.svg) |
![-\sqrt{\frac{1}{3}}\!\,](../I/m/802a5396c419b2085d70a3f18781a2d98ce619ce.svg) |
-1/2, 1 |
![\sqrt{\frac{3}{10}}\!\,](../I/m/a8f3246c01e59503eef68a1bf70a64808f198561.svg) |
![-\sqrt{\frac{8}{15}}\!\,](../I/m/175dd36040f520cc96c29c26fd6600c2d0ff0e24.svg) |
![\sqrt{\frac{1}{6}}\!\,](../I/m/8bdd5bf23f86b779473a724e85c90edef5a13f23.svg) |
|
j1=3/2, j2=3/2
m=3 |
j |
m1, m2 |
|
3 |
3/2, 3/2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
m=1 |
j |
m1, m2 |
|
3 |
2 |
1 |
3/2, -1/2 |
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
![\sqrt{\frac{1}{2}}\!\,](../I/m/af39f634eb891de4573e5b2e486b13a6d2df7ea4.svg) |
![\sqrt{\frac{3}{10}}\!\,](../I/m/a8f3246c01e59503eef68a1bf70a64808f198561.svg) |
1/2, 1/2 |
![\sqrt{\frac{3}{5}}\!\,](../I/m/a3c6237454cec1941476de55a9b34e4b3cbbd843.svg) |
![0\!\,](../I/m/0e2edcd1faf59bea6ce54ddee549c61963898857.svg) |
![-\sqrt{\frac{2}{5}}\!\,](../I/m/505afaf8f019cfc7dc32d7cbd144225caa7aec14.svg) |
-1/2, 3/2 |
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
![-\sqrt{\frac{1}{2}}\!\,](../I/m/46bcff57e7b0c158962ed97e3bfd834d79e07e9e.svg) |
![\sqrt{\frac{3}{10}}\!\,](../I/m/a8f3246c01e59503eef68a1bf70a64808f198561.svg) |
|
m=0 |
j |
m1, m2 |
|
3 |
2 |
1 |
0 |
3/2, -3/2 |
![\sqrt{\frac{1}{20}}\!\,](../I/m/e6288510bf0d601285da567ef6d61940e8f6ef64.svg) |
![\frac{1}{2}\!\,](../I/m/26a4cc810af1a06e12ab1f31375d6999d9d135ff.svg) |
![\sqrt{\frac{9}{20}}\!\,](../I/m/44588497a771377e28aa03e5b5245098a6b2010c.svg) |
![\frac{1}{2}\!\,](../I/m/26a4cc810af1a06e12ab1f31375d6999d9d135ff.svg) |
1/2, -1/2 |
![\sqrt{\frac{9}{20}}\!\,](../I/m/44588497a771377e28aa03e5b5245098a6b2010c.svg) |
![\frac{1}{2}\!\,](../I/m/26a4cc810af1a06e12ab1f31375d6999d9d135ff.svg) |
![-\sqrt{\frac{1}{20}}\!\,](../I/m/637ea15319edf10a8bd7eed0a0586520d1c4d078.svg) |
![-\frac{1}{2}\!\,](../I/m/0b315d7de5294fab6258a07775d5905697e8fae5.svg) |
-1/2, 1/2 |
![\sqrt{\frac{9}{20}}\!\,](../I/m/44588497a771377e28aa03e5b5245098a6b2010c.svg) |
![-\frac{1}{2}\!\,](../I/m/0b315d7de5294fab6258a07775d5905697e8fae5.svg) |
![-\sqrt{\frac{1}{20}}\!\,](../I/m/637ea15319edf10a8bd7eed0a0586520d1c4d078.svg) |
![\frac{1}{2}\!\,](../I/m/26a4cc810af1a06e12ab1f31375d6999d9d135ff.svg) |
-3/2, 3/2 |
![\sqrt{\frac{1}{20}}\!\,](../I/m/e6288510bf0d601285da567ef6d61940e8f6ef64.svg) |
![-\frac{1}{2}\!\,](../I/m/0b315d7de5294fab6258a07775d5905697e8fae5.svg) |
![\sqrt{\frac{9}{20}}\!\,](../I/m/44588497a771377e28aa03e5b5245098a6b2010c.svg) |
![-\frac{1}{2}\!\,](../I/m/0b315d7de5294fab6258a07775d5905697e8fae5.svg) |
|
j1=2, j2=1/2
m=5/2 |
j |
m1, m2 |
|
5/2 |
2, 1/2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
j1=2, j2=1
m=3 |
j |
m1, m2 |
|
3 |
2, 1 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
j1=2, j2=3/2
m=7/2 |
j |
m1, m2 |
|
7/2 |
2, 3/2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
m=3/2 |
j |
m1, m2 |
|
7/2 |
5/2 |
3/2 |
2, -1/2 |
![\sqrt{\frac{1}{7}}\!\,](../I/m/6b919c5938a9b570fde405b10593fb448e70f84e.svg) |
![\sqrt{\frac{16}{35}}\!\,](../I/m/5c4fe9ec192636d38418edf77d1aa115d2c9bf9c.svg) |
![\sqrt{\frac{2}{5}}\!\,](../I/m/7fea8a7ece9a47b8a32a9f4f75cbffc027f810bc.svg) |
1, 1/2 |
![\sqrt{\frac{4}{7}}\!\,](../I/m/b9816980a7d7a59c65030aeaec1a810de5b66046.svg) |
![\sqrt{\frac{1}{35}}\!\,](../I/m/1eca42ec294e34a20572fe2669db615e088c8ebd.svg) |
![-\sqrt{\frac{2}{5}}\!\,](../I/m/505afaf8f019cfc7dc32d7cbd144225caa7aec14.svg) |
0, 3/2 |
![\sqrt{\frac{2}{7}}\!\,](../I/m/55dee573f329129e9e2e4c150f277cd19b4a2f76.svg) |
![-\sqrt{\frac{18}{35}}\!\,](../I/m/1d4d379e56c800bb2b78c0230206a06638c82582.svg) |
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
|
m=1/2 |
j |
m1, m2 |
|
7/2 |
5/2 |
3/2 |
1/2 |
2, -3/2 |
![\sqrt{\frac{1}{35}}\!\,](../I/m/1eca42ec294e34a20572fe2669db615e088c8ebd.svg) |
![\sqrt{\frac{6}{35}}\!\,](../I/m/a441c9de6bdcb28fb19ad57848657a61e068d75d.svg) |
|
![\sqrt{\frac{2}{5}}\!\,](../I/m/7fea8a7ece9a47b8a32a9f4f75cbffc027f810bc.svg) |
1, -1/2 |
![\sqrt{\frac{12}{35}}\!\,](../I/m/da8be7435a3d372596ee016210fe3de55f604042.svg) |
![\sqrt{\frac{5}{14}}\!\,](../I/m/3f47f109c5e19615d811ff7a71404395de94b8e1.svg) |
![0\!\,](../I/m/0e2edcd1faf59bea6ce54ddee549c61963898857.svg) |
![-\sqrt{\frac{3}{10}}\!\,](../I/m/432b4023144e8d3135daa70e8263a01cda25d96f.svg) |
0, 1/2 |
![\sqrt{\frac{18}{35}}\!\,](../I/m/ba762b186bc838d24a8247804961cc2efef42d0e.svg) |
![-\sqrt{\frac{3}{35}}\!\,](../I/m/daf2ea6e71a583c33d15a721b6d255d78f3ed98d.svg) |
![-\sqrt{\frac{1}{5}}\!\,](../I/m/a394cb1e7a7fc66a7f2bdb8fa42be4235fb29d47.svg) |
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
-1, 3/2 |
![\sqrt{\frac{4}{35}}\!\,](../I/m/338591b750e99112d34eb2d9ad13fe4dea9527ce.svg) |
![-\sqrt{\frac{27}{70}}\!\,](../I/m/8d999827a79053c083a062be49cc788be9cf217b.svg) |
![\sqrt{\frac{2}{5}}\!\,](../I/m/7fea8a7ece9a47b8a32a9f4f75cbffc027f810bc.svg) |
![-\sqrt{\frac{1}{10}}\!\,](../I/m/2a9d612d346363370cd22e76424ce34dbea58ad8.svg) |
|
j1=2, j2=2
m=4 |
j |
m1, m2 |
|
4 |
2, 2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
m=1 |
j |
m1, m2 |
|
4 |
3 |
2 |
1 |
2, -1 |
![\sqrt{\frac{1}{14}}\!\,](../I/m/fd3d63b5979ac452a7e5243b55d0570d23a9db1f.svg) |
![\sqrt{\frac{3}{10}}\!\,](../I/m/a8f3246c01e59503eef68a1bf70a64808f198561.svg) |
|
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
1, 0 |
![\sqrt{\frac{3}{7}}\!\,](../I/m/126e7a8f05a8c377059060e56a4a281e467e2531.svg) |
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
![-\sqrt{\frac{1}{14}}\!\,](../I/m/9e4bd30e54575f26c20bfb5b99d1eab6a1743530.svg) |
![-\sqrt{\frac{3}{10}}\!\,](../I/m/432b4023144e8d3135daa70e8263a01cda25d96f.svg) |
0, 1 |
![\sqrt{\frac{3}{7}}\!\,](../I/m/126e7a8f05a8c377059060e56a4a281e467e2531.svg) |
![-\sqrt{\frac{1}{5}}\!\,](../I/m/a394cb1e7a7fc66a7f2bdb8fa42be4235fb29d47.svg) |
![-\sqrt{\frac{1}{14}}\!\,](../I/m/9e4bd30e54575f26c20bfb5b99d1eab6a1743530.svg) |
![\sqrt{\frac{3}{10}}\!\,](../I/m/a8f3246c01e59503eef68a1bf70a64808f198561.svg) |
-1, 2 |
![\sqrt{\frac{1}{14}}\!\,](../I/m/fd3d63b5979ac452a7e5243b55d0570d23a9db1f.svg) |
![-\sqrt{\frac{3}{10}}\!\,](../I/m/432b4023144e8d3135daa70e8263a01cda25d96f.svg) |
![\sqrt{\frac{3}{7}}\!\,](../I/m/126e7a8f05a8c377059060e56a4a281e467e2531.svg) |
![-\sqrt{\frac{1}{5}}\!\,](../I/m/a394cb1e7a7fc66a7f2bdb8fa42be4235fb29d47.svg) |
|
m=0 |
j |
m1, m2 |
|
4 |
3 |
2 |
1 |
0 |
2, -2 |
![\sqrt{\frac{1}{70}}\!\,](../I/m/04f05240fe279a52639045a851af77b7f7e497a0.svg) |
![\sqrt{\frac{1}{10}}\!\,](../I/m/e461f3044f6531ac49afc67501bcc70e1ef20051.svg) |
|
![\sqrt{\frac{2}{5}}\!\,](../I/m/7fea8a7ece9a47b8a32a9f4f75cbffc027f810bc.svg) |
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
1, -1 |
![\sqrt{\frac{8}{35}}\!\,](../I/m/525d7cdd6522bf6933bf293aeb4fb926b9b56216.svg) |
![\sqrt{\frac{2}{5}}\!\,](../I/m/7fea8a7ece9a47b8a32a9f4f75cbffc027f810bc.svg) |
![\sqrt{\frac{1}{14}}\!\,](../I/m/fd3d63b5979ac452a7e5243b55d0570d23a9db1f.svg) |
![-\sqrt{\frac{1}{10}}\!\,](../I/m/2a9d612d346363370cd22e76424ce34dbea58ad8.svg) |
![-\sqrt{\frac{1}{5}}\!\,](../I/m/a394cb1e7a7fc66a7f2bdb8fa42be4235fb29d47.svg) |
0, 0 |
![\sqrt{\frac{18}{35}}\!\,](../I/m/ba762b186bc838d24a8247804961cc2efef42d0e.svg) |
![0\!\,](../I/m/0e2edcd1faf59bea6ce54ddee549c61963898857.svg) |
![-\sqrt{\frac{2}{7}}\!\,](../I/m/e61371abb74baf45697f32cbfe75ce295b101eab.svg) |
|
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
-1, 1 |
![\sqrt{\frac{8}{35}}\!\,](../I/m/525d7cdd6522bf6933bf293aeb4fb926b9b56216.svg) |
![-\sqrt{\frac{2}{5}}\!\,](../I/m/505afaf8f019cfc7dc32d7cbd144225caa7aec14.svg) |
![\sqrt{\frac{1}{14}}\!\,](../I/m/fd3d63b5979ac452a7e5243b55d0570d23a9db1f.svg) |
![\sqrt{\frac{1}{10}}\!\,](../I/m/e461f3044f6531ac49afc67501bcc70e1ef20051.svg) |
![-\sqrt{\frac{1}{5}}\!\,](../I/m/a394cb1e7a7fc66a7f2bdb8fa42be4235fb29d47.svg) |
-2, 2 |
![\sqrt{\frac{1}{70}}\!\,](../I/m/04f05240fe279a52639045a851af77b7f7e497a0.svg) |
![-\sqrt{\frac{1}{10}}\!\,](../I/m/2a9d612d346363370cd22e76424ce34dbea58ad8.svg) |
![\sqrt{\frac{2}{7}}\!\,](../I/m/55dee573f329129e9e2e4c150f277cd19b4a2f76.svg) |
![-\sqrt{\frac{2}{5}}\!\,](../I/m/505afaf8f019cfc7dc32d7cbd144225caa7aec14.svg) |
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
|
j1=5/2, j2=1/2
m=3 |
j |
m1, m2 |
|
3 |
5/2, 1/2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
j1=5/2, j2=1
m=7/2 |
j |
m1, m2 |
|
7/2 |
5/2, 1 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
m=3/2 |
j |
m1, m2 |
|
7/2 |
5/2 |
3/2 |
5/2, -1 |
![\sqrt{\frac{1}{21}}\!\,](../I/m/b7d76303672b0cf2ea2b754dc79ed607d6ba9e46.svg) |
|
![\sqrt{\frac{2}{3}}\!\,](../I/m/3c82f3c071d6d88dcfef2195a56f3f5c85676c3d.svg) |
3/2, 0 |
![\sqrt{\frac{10}{21}}\!\,](../I/m/c0457b20e909baefc5d24ddab7f429549b2d12d0.svg) |
![\sqrt{\frac{9}{35}}\!\,](../I/m/b6cee30e22368d0b44343916c0d9a1ec243e3655.svg) |
![-\sqrt{\frac{4}{15}}\!\,](../I/m/5d5a98e2b050e933c4e899a4cd7e487e5515848c.svg) |
1/2, 1 |
![\sqrt{\frac{10}{21}}\!\,](../I/m/c0457b20e909baefc5d24ddab7f429549b2d12d0.svg) |
![-\sqrt{\frac{16}{35}}\!\,](../I/m/bca0a591c954f2e680b89ad474f893d4978a74db.svg) |
![\sqrt{\frac{1}{15}}\!\,](../I/m/dee88493dbf645c6f4079065117665d8f93ba578.svg) |
|
m=1/2 |
j |
m1, m2 |
|
7/2 |
5/2 |
3/2 |
3/2, -1 |
![\sqrt{\frac{1}{7}}\!\,](../I/m/6b919c5938a9b570fde405b10593fb448e70f84e.svg) |
![\sqrt{\frac{16}{35}}\!\,](../I/m/5c4fe9ec192636d38418edf77d1aa115d2c9bf9c.svg) |
![\sqrt{\frac{2}{5}}\!\,](../I/m/7fea8a7ece9a47b8a32a9f4f75cbffc027f810bc.svg) |
1/2, 0 |
![\sqrt{\frac{4}{7}}\!\,](../I/m/b9816980a7d7a59c65030aeaec1a810de5b66046.svg) |
![\sqrt{\frac{1}{35}}\!\,](../I/m/1eca42ec294e34a20572fe2669db615e088c8ebd.svg) |
![-\sqrt{\frac{2}{5}}\!\,](../I/m/505afaf8f019cfc7dc32d7cbd144225caa7aec14.svg) |
-1/2, 1 |
![\sqrt{\frac{2}{7}}\!\,](../I/m/55dee573f329129e9e2e4c150f277cd19b4a2f76.svg) |
![-\sqrt{\frac{18}{35}}\!\,](../I/m/1d4d379e56c800bb2b78c0230206a06638c82582.svg) |
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
|
j1=5/2, j2=3/2
m=4 |
j |
m1, m2 |
|
4 |
5/2, 3/2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
m=2 |
j |
m1, m2 |
|
4 |
3 |
2 |
5/2, -1/2 |
![\sqrt{\frac{3}{28}}\!\,](../I/m/dc15be7e29c03cf1ca234761d6a5a39eee063d9c.svg) |
![\sqrt{\frac{5}{12}}\!\,](../I/m/913f2109ec6111234a4a3bc22b44c49958bbb023.svg) |
![\sqrt{\frac{10}{21}}\!\,](../I/m/c0457b20e909baefc5d24ddab7f429549b2d12d0.svg) |
3/2, 1/2 |
![\sqrt{\frac{15}{28}}\!\,](../I/m/ca7b34b034fad0c66159cd1874dea3539761d02f.svg) |
![\sqrt{\frac{1}{12}}\!\,](../I/m/33f092414d1691e9a5fc275bc892e45dcc95826b.svg) |
![-\sqrt{\frac{8}{21}}\!\,](../I/m/6d98962db14d51baa50457ef1ec333b4904dad23.svg) |
1/2, 3/2 |
![\sqrt{\frac{5}{14}}\!\,](../I/m/3f47f109c5e19615d811ff7a71404395de94b8e1.svg) |
![-\sqrt{\frac{1}{2}}\!\,](../I/m/46bcff57e7b0c158962ed97e3bfd834d79e07e9e.svg) |
![\sqrt{\frac{1}{7}}\!\,](../I/m/6b919c5938a9b570fde405b10593fb448e70f84e.svg) |
|
m=1 |
j |
m1, m2 |
|
4 |
3 |
2 |
1 |
5/2, -3/2 |
![\sqrt{\frac{1}{56}}\!\,](../I/m/0d5cc0a95250c698953851e67218408b4aa47d2d.svg) |
![\sqrt{\frac{1}{8}}\!\,](../I/m/b92628aac75a2774adf2a7654018a8ab71868ab3.svg) |
![\sqrt{\frac{5}{14}}\!\,](../I/m/3f47f109c5e19615d811ff7a71404395de94b8e1.svg) |
![\sqrt{\frac{1}{2}}\!\,](../I/m/af39f634eb891de4573e5b2e486b13a6d2df7ea4.svg) |
3/2, -1/2 |
![\sqrt{\frac{15}{56}}\!\,](../I/m/99f734113d4bd4b3727cec9ebe37c57db4003319.svg) |
![\sqrt{\frac{49}{120}}\!\,](../I/m/826db4c9ec32cfb3e6b3b5d00728aa02296c948b.svg) |
![\sqrt{\frac{1}{42}}\!\,](../I/m/3ae2b326b33207bfd1336a041bc1c75cfb7746d4.svg) |
![-\sqrt{\frac{3}{10}}\!\,](../I/m/432b4023144e8d3135daa70e8263a01cda25d96f.svg) |
1/2, 1/2 |
![\sqrt{\frac{15}{28}}\!\,](../I/m/ca7b34b034fad0c66159cd1874dea3539761d02f.svg) |
![-\sqrt{\frac{1}{60}}\!\,](../I/m/88fd157ae76efc095c0dfac64a58fb0a7dfc5634.svg) |
![-\sqrt{\frac{25}{84}}\!\,](../I/m/9bd2613ff429cf4a4063eabacc27de4b5d80b100.svg) |
![\sqrt{\frac{3}{20}}\!\,](../I/m/bbac756c0b95b3cced301b2065ca9e53837d660f.svg) |
-1/2, 3/2 |
![\sqrt{\frac{5}{28}}\!\,](../I/m/aa49602770b2034563b8d05f63d1c2af44f43a56.svg) |
![-\sqrt{\frac{9}{20}}\!\,](../I/m/8c36d8e12882b3a84a61bb8e374f5f1a46c6aed2.svg) |
![\sqrt{\frac{9}{28}}\!\,](../I/m/52696dc1ae277d2ec2ef8e8417e0e2d59cb10fbb.svg) |
![-\sqrt{\frac{1}{20}}\!\,](../I/m/637ea15319edf10a8bd7eed0a0586520d1c4d078.svg) |
|
m=0 |
j |
m1, m2 |
|
4 |
3 |
2 |
1 |
3/2, -3/2 |
![\sqrt{\frac{1}{14}}\!\,](../I/m/fd3d63b5979ac452a7e5243b55d0570d23a9db1f.svg) |
![\sqrt{\frac{3}{10}}\!\,](../I/m/a8f3246c01e59503eef68a1bf70a64808f198561.svg) |
|
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
1/2, -1/2 |
![\sqrt{\frac{3}{7}}\!\,](../I/m/126e7a8f05a8c377059060e56a4a281e467e2531.svg) |
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
![-\sqrt{\frac{1}{14}}\!\,](../I/m/9e4bd30e54575f26c20bfb5b99d1eab6a1743530.svg) |
![-\sqrt{\frac{3}{10}}\!\,](../I/m/432b4023144e8d3135daa70e8263a01cda25d96f.svg) |
-1/2, 1/2 |
![\sqrt{\frac{3}{7}}\!\,](../I/m/126e7a8f05a8c377059060e56a4a281e467e2531.svg) |
![-\sqrt{\frac{1}{5}}\!\,](../I/m/a394cb1e7a7fc66a7f2bdb8fa42be4235fb29d47.svg) |
![-\sqrt{\frac{1}{14}}\!\,](../I/m/9e4bd30e54575f26c20bfb5b99d1eab6a1743530.svg) |
![\sqrt{\frac{3}{10}}\!\,](../I/m/a8f3246c01e59503eef68a1bf70a64808f198561.svg) |
-3/2, 3/2 |
![\sqrt{\frac{1}{14}}\!\,](../I/m/fd3d63b5979ac452a7e5243b55d0570d23a9db1f.svg) |
![-\sqrt{\frac{3}{10}}\!\,](../I/m/432b4023144e8d3135daa70e8263a01cda25d96f.svg) |
![\sqrt{\frac{3}{7}}\!\,](../I/m/126e7a8f05a8c377059060e56a4a281e467e2531.svg) |
![-\sqrt{\frac{1}{5}}\!\,](../I/m/a394cb1e7a7fc66a7f2bdb8fa42be4235fb29d47.svg) |
|
j1=5/2, j2=2
m=9/2 |
j |
m1, m2 |
|
9/2 |
5/2, 2 |
![1\!\,](../I/m/d03b3afac31e6ed4f4a55aeb4736b568ebf715f9.svg) |
|
m=5/2 |
j |
m1, m2 |
|
9/2 |
7/2 |
5/2 |
5/2, 0 |
![\sqrt{\frac{1}{6}}\!\,](../I/m/8bdd5bf23f86b779473a724e85c90edef5a13f23.svg) |
![\sqrt{\frac{10}{21}}\!\,](../I/m/c0457b20e909baefc5d24ddab7f429549b2d12d0.svg) |
![\sqrt{\frac{5}{14}}\!\,](../I/m/3f47f109c5e19615d811ff7a71404395de94b8e1.svg) |
3/2, 1 |
![\sqrt{\frac{5}{9}}\!\,](../I/m/cc6d00be1de2447fa5d0feac8e1f3786f4f2aea8.svg) |
![\sqrt{\frac{1}{63}}\!\,](../I/m/e17f5b995cc1904862b811077abc219ade7722e1.svg) |
![-\sqrt{\frac{3}{7}}\!\,](../I/m/28a555f44a97885299832260d3efb2a9c41e8920.svg) |
1/2, 2 |
![\sqrt{\frac{5}{18}}\!\,](../I/m/e6dad3f0f0443132adbca4cfc47a96df336e5250.svg) |
![-\sqrt{\frac{32}{63}}\!\,](../I/m/cdaf0bfd5599bfe7865d13bba64ebacf1bb5ffaa.svg) |
![\sqrt{\frac{3}{14}}\!\,](../I/m/dd801f526b93d237237b3f32d35d7c5895b7d491.svg) |
|
m=3/2 |
j |
m1, m2 |
|
9/2 |
7/2 |
5/2 |
3/2 |
5/2, -1 |
![\sqrt{\frac{1}{21}}\!\,](../I/m/b7d76303672b0cf2ea2b754dc79ed607d6ba9e46.svg) |
![\sqrt{\frac{5}{21}}\!\,](../I/m/9525035fc97179ae127b459978532fb6f7ea8070.svg) |
|
![\sqrt{\frac{2}{7}}\!\,](../I/m/55dee573f329129e9e2e4c150f277cd19b4a2f76.svg) |
3/2, 0 |
![\sqrt{\frac{5}{14}}\!\,](../I/m/3f47f109c5e19615d811ff7a71404395de94b8e1.svg) |
![\sqrt{\frac{2}{7}}\!\,](../I/m/55dee573f329129e9e2e4c150f277cd19b4a2f76.svg) |
![-\sqrt{\frac{1}{70}}\!\,](../I/m/1c74d71ec37ac2bdaeb8a19c4b3a3bfc183e2ecf.svg) |
![-\sqrt{\frac{12}{35}}\!\,](../I/m/026ffa5be786a3b0b9e43cdf67eb9033afdb9b87.svg) |
1/2, 1 |
![\sqrt{\frac{10}{21}}\!\,](../I/m/c0457b20e909baefc5d24ddab7f429549b2d12d0.svg) |
![-\sqrt{\frac{2}{21}}\!\,](../I/m/f5908819c4d768a6fafde5e10d1d47d84a3136b7.svg) |
![-\sqrt{\frac{6}{35}}\!\,](../I/m/944d168ff29ba30cb932e5e1e598674fad466139.svg) |
![\sqrt{\frac{9}{35}}\!\,](../I/m/b6cee30e22368d0b44343916c0d9a1ec243e3655.svg) |
-1/2, 2 |
![\sqrt{\frac{5}{42}}\!\,](../I/m/a0c77456639edf3336d2fb6cbb915182da0af748.svg) |
![-\sqrt{\frac{8}{21}}\!\,](../I/m/6d98962db14d51baa50457ef1ec333b4904dad23.svg) |
![\sqrt{\frac{27}{70}}\!\,](../I/m/44286962a672ea0a75cd07bbe14972cd72d8fd07.svg) |
![-\sqrt{\frac{4}{35}}\!\,](../I/m/b4512079bda6ac2f9dc38648d4b39ae0c1585551.svg) |
|
m=1/2 |
j |
m1, m2 |
|
9/2 |
7/2 |
5/2 |
3/2 |
1/2 |
5/2, -2 |
![\sqrt{\frac{1}{126}}\!\,](../I/m/b7b67ae0570754ad07be554ee4c428080e3eb301.svg) |
![\sqrt{\frac{4}{63}}\!\,](../I/m/06a1f9037d32c2bd6c7e4650c5135bc43a6883dc.svg) |
![\sqrt{\frac{3}{14}}\!\,](../I/m/dd801f526b93d237237b3f32d35d7c5895b7d491.svg) |
![\sqrt{\frac{8}{21}}\!\,](../I/m/2bbbdfaf486aa58b7005d63a2c5ad051d04527c9.svg) |
![\sqrt{\frac{1}{3}}\!\,](../I/m/8278116272f781236495e1c288e57bf02fed53da.svg) |
3/2, -1 |
![\sqrt{\frac{10}{63}}\!\,](../I/m/5e2908c85e0816b860f0322681cee3bee4c8f18d.svg) |
![\sqrt{\frac{121}{315}}\!\,](../I/m/cc7a9cb86bf02306ce711c2b0d238762ebd96319.svg) |
![\sqrt{\frac{6}{35}}\!\,](../I/m/a441c9de6bdcb28fb19ad57848657a61e068d75d.svg) |
![-\sqrt{\frac{2}{105}}\!\,](../I/m/e78dd710b0a7f46305d16df19c534e069e0f42d9.svg) |
![-\sqrt{\frac{4}{15}}\!\,](../I/m/5d5a98e2b050e933c4e899a4cd7e487e5515848c.svg) |
1/2, 0 |
![\sqrt{\frac{10}{21}}\!\,](../I/m/c0457b20e909baefc5d24ddab7f429549b2d12d0.svg) |
![\sqrt{\frac{4}{105}}\!\,](../I/m/8bb3aa836b9de4811c06f1f1033d9be58b45f3c7.svg) |
![-\sqrt{\frac{8}{35}}\!\,](../I/m/8ac43fad34de3cb6479bd8a73c838263b6927853.svg) |
![-\sqrt{\frac{2}{35}}\!\,](../I/m/a225e5a2ee2bd1109be6c95653161c7542542364.svg) |
![\sqrt{\frac{1}{5}}\!\,](../I/m/b951ecd7ef3e2fc31ec9ba06a664c10234f199d2.svg) |
-1/2, 1 |
![\sqrt{\frac{20}{63}}\!\,](../I/m/e08fc7bbc8f8054db140dc1a448c1829af357d7a.svg) |
![-\sqrt{\frac{14}{45}}\!\,](../I/m/3e8cc8383ec54e6407c4cabe5e1c9ec8c135b2ad.svg) |
![0\!\,](../I/m/0e2edcd1faf59bea6ce54ddee549c61963898857.svg) |
![\sqrt{\frac{5}{21}}\!\,](../I/m/9525035fc97179ae127b459978532fb6f7ea8070.svg) |
![-\sqrt{\frac{2}{15}}\!\,](../I/m/bc92c37c6fe14d2858bf88df48bf1ea951be59e9.svg) |
-3/2, 2 |
![\sqrt{\frac{5}{126}}\!\,](../I/m/ec85891524b61cf429bd5f1a06e6d187707f23c2.svg) |
![-\sqrt{\frac{64}{315}}\!\,](../I/m/900a99594e459b90d408971140daeeab3aab2aac.svg) |
![\sqrt{\frac{27}{70}}\!\,](../I/m/44286962a672ea0a75cd07bbe14972cd72d8fd07.svg) |
![-\sqrt{\frac{32}{105}}\!\,](../I/m/3269a739a44e81a4866668431810cc2665f49976.svg) |
![\sqrt{\frac{1}{15}}\!\,](../I/m/dee88493dbf645c6f4079065117665d8f93ba578.svg) |
|
SU(N) Clebsch–Gordan coefficients
Algorithms to produce Clebsch–Gordan coefficients for higher values of
and
, or for the su(N) algebra instead of su(2), are known.[6]
A web interface for tabulating SU(N) Clebsch-Gordan coefficients is readily available.
References
- ↑ Baird, C.E.; L. C. Biedenharn (October 1964). "On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SUn". J. Math. Phys. 5: 1723–1730. Bibcode:1964JMP.....5.1723B. doi:10.1063/1.1704095. Retrieved 2007-12-20.
- ↑ Hagiwara, K.; et al. (July 2002). "Review of Particle Properties" (PDF). Phys. Rev. D. 66: 010001. Bibcode:2002PhRvD..66a0001H. doi:10.1103/PhysRevD.66.010001. Retrieved 2007-12-20.
- ↑ Mathar, Richard J. (2006-08-14). "SO(3) Clebsch Gordan coefficients" (text). Retrieved 2012-10-15.
- ↑ (2.41), p. 172, Quantum Mechanics: Foundations and Applications, Arno Bohm, M. Loewe, New York: Springer-Verlag, 3rd ed., 1993, ISBN 0-387-95330-2.
- ↑ Weisbluth, Michael (1978). Atoms and molecules. ACADEMIC PRESS. p. 28. ISBN 0-12-744450-5. Table 1.4 resumes the most common.
- ↑ Alex, A.; M. Kalus; A. Huckleberry; J. von Delft (February 2011). "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients". J. Math. Phys. 82: 023507. arXiv:1009.0437
. Bibcode:2011JMP....52b3507A. doi:10.1063/1.3521562. Retrieved 2011-04-13.
External links