Universal Transverse Mercator coordinate system: Difference between revisions

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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 <math>j_1</math>, <math>j_2</math>, <math>j</math> is arbitrary to some degree and has been fixed according to the Condon-Shortley and Wigner sign convention as discussed by Baird and [[Lawrence Biedenharn|Biedenharn]].<ref>{{cite journal |last=Baird |first=C.E. |coauthors=L. C. Biedenharn |title=On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SU<sub>n</sub> |journal=J. Math. Phys. |volume=5 |date=October 1964 |pages=1723–1730 |doi=10.1063/1.1704095 |url=http://link.aip.org/link/?JMAPAQ/5/1723/1 |accessdate=2007-12-20 |bibcode=1964JMP.....5.1723B}}</ref>  Tables with the same sign convention may be found in the [[Particle Data Group]]'s ''Review of Particle Properties''<ref>{{cite journal |last=Hagiwara |first=K. |coauthors=''et al.'' |title=Review of Particle Properties |journal=Phys. Rev. D |volume=66 |date=July 2002 |pages=010001 |doi=10.1103/PhysRevD.66.010001 |url=http://pdg.lbl.gov/2002/clebrpp.pdf |format=PDF |accessdate=2007-12-20 |bibcode=2002PhRvD..66a0001H}}</ref> and in online tables.<ref>{{cite web |last=Mathar |first=Richard J. |title=SO(3) Clebsch Gordan coefficients |date=2006-08-14 |url=http://www.mpia.de/~mathar/progs/CGord |format=text |accessdate=2012-10-15}}</ref>
 
==Formulation==
The Clebsch-Gordan coefficients are the solutions to
 
<math>
  |(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
</math>
 
Explicitly:
 
<math>\langle j_1j_2;m_1m_2|j_1j_2;jm\rangle=</math>
 
<math>\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
</math>
 
<math>
\sqrt{(j+m)!(j-m)!(j_1-m_1)!(j_1+m_1)!(j_2-m_2)!(j_2+m_2)!}\ \times
</math>
 
<math>
\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)!}.
</math>
 
The summation is extended over all integer ''k'' for which the argument of every factorial is nonnegative.<ref>(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.</ref>
 
For brevity, solutions with m < 0 and j<sub>1</sub> < j<sub>2</sub> are omitted.  They may be calculated using the simple relations
 
:<math>\langle j_1j_2;m_1m_2|j_1j_2;jm\rangle=(-1)^{j-j_1-j_2}\langle j_1j_2;-m_1,-m_2|j_1j_2;j,-m\rangle</math> .
 
and
 
:<math>\langle j_1j_2;m_1m_2|j_1j_2;jm\rangle=(-1)^{j-j_1-j_2} \langle j_2j_1;m_2m_1|j_2j_1;jm\rangle</math> .
 
A complete list<ref>{{cite book|last=Weisbluth|first=Michael|title=Atoms and molecules|year=1978|publisher=ACADEMIC PRESS|isbn=0-12-744450-5|page=28}} Table 1.4 resumes the most common.</ref>
 
===j<sub>2</sub>=0===
 
When j<sub>2</sub> = 0, the Clebsch-Gordan coefficients are given by <math>\delta_{j,j_1}\delta_{m,m_1}</math> .
 
===j<sub>1</sub>=1/2, j<sub>2</sub>=1/2===
 
{|
|----- align="center"
| m=1 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''1'''
|----- align="center"
| '''1/2,&nbsp;1/2''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=0 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
  || '''1''' || '''0'''
|----- align="center"
| '''1/2,&nbsp;-1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=1, j<sub>2</sub>=1/2===
 
{|
|----- align="center"
| m=3/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3/2'''
|----- align="center"
| '''1,&nbsp;1/2''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3/2''' || '''1/2'''
|----- align="center"
| '''1,&nbsp;-1/2''' || <math>\sqrt{\frac{1}{3}}\!\,</math>
| <math>\sqrt{\frac{2}{3}}\!\,</math>
|----- align="center"
| '''0,&nbsp;1/2''' || <math>\sqrt{\frac{2}{3}}\!\,</math>
| <math>-\sqrt{\frac{1}{3}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=1, j<sub>2</sub>=1===
 
{|
|----- align="center"
| m=2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''2'''
|----- align="center"
| '''1,&nbsp;1''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''2''' || '''1'''
|----- align="center"
| '''1,&nbsp;0''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math>
|----- align="center"
| '''0,&nbsp;1''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=0 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''2''' || '''1''' || '''0'''
|----- align="center"
| '''1,&nbsp;-1''' || <math>\sqrt{\frac{1}{6}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math> || <math>\sqrt{\frac{1}{3}}\!\,</math>
|----- align="center"
| '''0,&nbsp;0''' || <math>\sqrt{\frac{2}{3}}\!\,</math>
| <math>0\!\,</math>
| <math>-\sqrt{\frac{1}{3}}\!\,</math>
|----- align="center"
| '''-1,&nbsp;1''' || <math>\sqrt{\frac{1}{6}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{1}{3}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=3/2, j<sub>2</sub>=1/2===
 
{|
|----- align="center"
| m=2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''2'''
|----- align="center"
| '''3/2,&nbsp;1/2''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''2''' || '''1'''
|----- align="center"
| '''3/2,&nbsp;-1/2''' || <math>\frac{1}{2}\!\,</math>
| <math>\sqrt{\frac{3}{4}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;1/2''' || <math>\sqrt{\frac{3}{4}}\!\,</math>
| <math>-\frac{1}{2}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=0 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''2''' || '''1'''
|----- align="center"
| '''1/2,&nbsp;-1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=3/2, j<sub>2</sub>=1===
 
{|
|----- align="center"
| m=5/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''5/2'''
|----- align="center"
| '''3/2,&nbsp;1''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=3/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''5/2''' || '''3/2'''
|----- align="center"
| '''3/2,&nbsp;0''' || <math>\sqrt{\frac{2}{5}}\!\,</math>
| <math>\sqrt{\frac{3}{5}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;1''' || <math>\sqrt{\frac{3}{5}}\!\,</math>
| <math>-\sqrt{\frac{2}{5}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''5/2''' || '''3/2''' || '''1/2'''
|----- align="center"
| '''3/2,&nbsp;-1'''
| <math>\sqrt{\frac{1}{10}}\!\,</math>
| <math>\sqrt{\frac{2}{5}}\!\,</math> || <math>\sqrt{\frac{1}{2}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;0''' || <math>\sqrt{\frac{3}{5}}\!\,</math>
| <math>\sqrt{\frac{1}{15}}\!\,</math>
| <math>-\sqrt{\frac{1}{3}}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;1'''
| <math>\sqrt{\frac{3}{10}}\!\,</math>
| <math>-\sqrt{\frac{8}{15}}\!\,</math>
| <math>\sqrt{\frac{1}{6}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=3/2, j<sub>2</sub>=3/2===
 
{|
|----- align="center"
| m=3 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3'''
|----- align="center"
| '''3/2,&nbsp;3/2''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3''' || '''2'''
|----- align="center"
| '''3/2,&nbsp;1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;3/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3''' || '''2''' || '''1'''
|----- align="center"
| '''3/2,&nbsp;-1/2''' || <math>\sqrt{\frac{1}{5}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{3}{10}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;1/2''' || <math>\sqrt{\frac{3}{5}}\!\,</math>
| <math>0\!\,</math>
| <math>-\sqrt{\frac{2}{5}}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;3/2''' || <math>\sqrt{\frac{1}{5}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{3}{10}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=0 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3''' || '''2''' || '''1''' || '''0'''
|----- align="center"
| '''3/2,&nbsp;-3/2'''
| <math>\sqrt{\frac{1}{20}}\!\,</math>
| <math>\frac{1}{2}\!\,</math>
| <math>\sqrt{\frac{9}{20}}\!\,</math>
| <math>\frac{1}{2}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;-1/2'''
| <math>\sqrt{\frac{9}{20}}\!\,</math>
| <math>\frac{1}{2}\!\,</math>
| <math>-\sqrt{\frac{1}{20}}\!\,</math>
| <math>-\frac{1}{2}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;1/2'''
| <math>\sqrt{\frac{9}{20}}\!\,</math>
| <math>-\frac{1}{2}\!\,</math>
| <math>-\sqrt{\frac{1}{20}}\!\,</math>
| <math>\frac{1}{2}\!\,</math>
|----- align="center"
| '''-3/2,&nbsp;3/2'''
| <math>\sqrt{\frac{1}{20}}\!\,</math>
| <math>-\frac{1}{2}\!\,</math>
| <math>\sqrt{\frac{9}{20}}\!\,</math>
| <math>-\frac{1}{2}\!\,</math>
|}
|}
 
===j<sub>1</sub>=2, j<sub>2</sub>=1/2===
 
{|
|----- align="center"
| m=5/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''5/2'''
|----- align="center"
| '''2,&nbsp;1/2''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=3/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''5/2''' || '''3/2'''
|----- align="center"
| '''2,&nbsp;-1/2''' || <math>\sqrt{\frac{1}{5}}\!\,</math>
| <math>\sqrt{\frac{4}{5}}\!\,</math>
|----- align="center"
| '''1,&nbsp;1/2''' || <math>\sqrt{\frac{4}{5}}\!\,</math>
| <math>-\sqrt{\frac{1}{5}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''5/2''' || '''3/2'''
|----- align="center"
| '''1,&nbsp;-1/2''' || <math>\sqrt{\frac{2}{5}}\!\,</math>
| <math>\sqrt{\frac{3}{5}}\!\,</math>
|----- align="center"
| '''0,&nbsp;1/2''' || <math>\sqrt{\frac{3}{5}}\!\,</math>
| <math>-\sqrt{\frac{2}{5}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=2, j<sub>2</sub>=1===
 
{|
|----- align="center"
| m=3 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3'''
|----- align="center"
| '''2,&nbsp;1''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3''' || '''2'''
|----- align="center"
| '''2,&nbsp;0''' || <math>\sqrt{\frac{1}{3}}\!\,</math>
| <math>\sqrt{\frac{2}{3}}\!\,</math>
|----- align="center"
| '''1,&nbsp;1''' || <math>\sqrt{\frac{2}{3}}\!\,</math>
| <math>-\sqrt{\frac{1}{3}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3''' || '''2''' || '''1'''
|----- align="center"
| '''2,&nbsp;-1'''
| <math>\sqrt{\frac{1}{15}}\!\,</math>
| <math>\sqrt{\frac{1}{3}}\!\,</math> || <math>\sqrt{\frac{3}{5}}\!\,</math>
|----- align="center"
| '''1,&nbsp;0'''
| <math>\sqrt{\frac{8}{15}}\!\,</math>
| <math>\sqrt{\frac{1}{6}}\!\,</math>
| <math>-\sqrt{\frac{3}{10}}\!\,</math>
|----- align="center"
| '''0,&nbsp;1''' || <math>\sqrt{\frac{2}{5}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{1}{10}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=0 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3''' || '''2''' || '''1'''
|----- align="center"
| '''1,&nbsp;-1''' || <math>\sqrt{\frac{1}{5}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{3}{10}}\!\,</math>
|----- align="center"
| '''0,&nbsp;0''' || <math>\sqrt{\frac{3}{5}}\!\,</math>
| <math>0\!\,</math>
| <math>-\sqrt{\frac{2}{5}}\!\,</math>
|----- align="center"
| '''-1,&nbsp;1''' || <math>\sqrt{\frac{1}{5}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{3}{10}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=2, j<sub>2</sub>=3/2===
 
{|
|----- align="center"
| m=7/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''7/2'''
|----- align="center"
| '''2,&nbsp;3/2''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=5/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''7/2''' || '''5/2'''
|----- align="center"
| '''2,&nbsp;1/2''' || <math>\sqrt{\frac{3}{7}}\!\,</math>
| <math>\sqrt{\frac{4}{7}}\!\,</math>
|----- align="center"
| '''1,&nbsp;3/2''' || <math>\sqrt{\frac{4}{7}}\!\,</math>
| <math>-\sqrt{\frac{3}{7}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=3/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''7/2''' || '''5/2''' || '''3/2'''
|----- align="center"
| '''2,&nbsp;-1/2''' || <math>\sqrt{\frac{1}{7}}\!\,</math>
| <math>\sqrt{\frac{16}{35}}\!\,</math>
| <math>\sqrt{\frac{2}{5}}\!\,</math>
|----- align="center"
| '''1,&nbsp;1/2''' || <math>\sqrt{\frac{4}{7}}\!\,</math>
| <math>\sqrt{\frac{1}{35}}\!\,</math>
| <math>-\sqrt{\frac{2}{5}}\!\,</math>
|----- align="center"
| '''0,&nbsp;3/2''' || <math>\sqrt{\frac{2}{7}}\!\,</math>
| <math>-\sqrt{\frac{18}{35}}\!\,</math>
| <math>\sqrt{\frac{1}{5}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''7/2''' || '''5/2''' || '''3/2''' || '''1/2'''
|----- align="center"
| '''2,&nbsp;-3/2'''
| <math>\sqrt{\frac{1}{35}}\!\,</math>
| <math>\sqrt{\frac{6}{35}}\!\,</math>
| <math>\sqrt{\frac{2}{5}}\!\,</math> || <math>\sqrt{\frac{2}{5}}\!\,</math>
|----- align="center"
| '''1,&nbsp;-1/2'''
| <math>\sqrt{\frac{12}{35}}\!\,</math>
| <math>\sqrt{\frac{5}{14}}\!\,</math>
| <math>0\!\,</math>
| <math>-\sqrt{\frac{3}{10}}\!\,</math>
|----- align="center"
| '''0,&nbsp;1/2'''
| <math>\sqrt{\frac{18}{35}}\!\,</math>
| <math>-\sqrt{\frac{3}{35}}\!\,</math>
| <math>-\sqrt{\frac{1}{5}}\!\,</math>
| <math>\sqrt{\frac{1}{5}}\!\,</math>
|----- align="center"
| '''-1,&nbsp;3/2'''
| <math>\sqrt{\frac{4}{35}}\!\,</math>
| <math>-\sqrt{\frac{27}{70}}\!\,</math>
| <math>\sqrt{\frac{2}{5}}\!\,</math>
| <math>-\sqrt{\frac{1}{10}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=2, j<sub>2</sub>=2===
 
{|
|----- align="center"
| m=4 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''4'''
|----- align="center"
| '''2,&nbsp;2''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=3 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''4''' || '''3'''
|----- align="center"
| '''2,&nbsp;1''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math>
|----- align="center"
| '''1,&nbsp;2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''4''' || '''3''' || '''2'''
|----- align="center"
| '''2,&nbsp;0'''
| <math>\sqrt{\frac{3}{14}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math> || <math>\sqrt{\frac{2}{7}}\!\,</math>
|----- align="center"
| '''1,&nbsp;1''' || <math>\sqrt{\frac{4}{7}}\!\,</math>
| <math>0\!\,</math>
| <math>-\sqrt{\frac{3}{7}}\!\,</math>
|----- align="center"
| '''0,&nbsp;2'''
| <math>\sqrt{\frac{3}{14}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{2}{7}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''4''' || '''3''' || '''2''' || '''1'''
|----- align="center"
| '''2,&nbsp;-1'''
| <math>\sqrt{\frac{1}{14}}\!\,</math>
| <math>\sqrt{\frac{3}{10}}\!\,</math>
| <math>\sqrt{\frac{3}{7}}\!\,</math> || <math>\sqrt{\frac{1}{5}}\!\,</math>
|----- align="center"
| '''1,&nbsp;0''' || <math>\sqrt{\frac{3}{7}}\!\,</math>
| <math>\sqrt{\frac{1}{5}}\!\,</math>
| <math>-\sqrt{\frac{1}{14}}\!\,</math>
| <math>-\sqrt{\frac{3}{10}}\!\,</math>
|----- align="center"
| '''0,&nbsp;1''' || <math>\sqrt{\frac{3}{7}}\!\,</math>
| <math>-\sqrt{\frac{1}{5}}\!\,</math>
| <math>-\sqrt{\frac{1}{14}}\!\,</math>
| <math>\sqrt{\frac{3}{10}}\!\,</math>
|----- align="center"
| '''-1,&nbsp;2'''
| <math>\sqrt{\frac{1}{14}}\!\,</math>
| <math>-\sqrt{\frac{3}{10}}\!\,</math>
| <math>\sqrt{\frac{3}{7}}\!\,</math>
| <math>-\sqrt{\frac{1}{5}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=0 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''4''' || '''3''' || '''2''' || '''1'''
| '''0'''
|----- align="center"
| '''2,&nbsp;-2'''
| <math>\sqrt{\frac{1}{70}}\!\,</math>
| <math>\sqrt{\frac{1}{10}}\!\,</math>
| <math>\sqrt{\frac{2}{7}}\!\,</math> || <math>\sqrt{\frac{2}{5}}\!\,</math>
| <math>\sqrt{\frac{1}{5}}\!\,</math>
|----- align="center"
| '''1,&nbsp;-1'''
| <math>\sqrt{\frac{8}{35}}\!\,</math>
| <math>\sqrt{\frac{2}{5}}\!\,</math>
| <math>\sqrt{\frac{1}{14}}\!\,</math>
| <math>-\sqrt{\frac{1}{10}}\!\,</math>
| <math>-\sqrt{\frac{1}{5}}\!\,</math>
|----- align="center"
| '''0,&nbsp;0'''
| <math>\sqrt{\frac{18}{35}}\!\,</math>
| <math>0\!\,</math>
| <math>-\sqrt{\frac{2}{7}}\!\,</math>
| <math>0\!\,</math> || <math>\sqrt{\frac{1}{5}}\!\,</math>
|----- align="center"
| '''-1,&nbsp;1'''
| <math>\sqrt{\frac{8}{35}}\!\,</math>
| <math>-\sqrt{\frac{2}{5}}\!\,</math>
| <math>\sqrt{\frac{1}{14}}\!\,</math>
| <math>\sqrt{\frac{1}{10}}\!\,</math>
| <math>-\sqrt{\frac{1}{5}}\!\,</math>
|----- align="center"
| '''-2,&nbsp;2'''
| <math>\sqrt{\frac{1}{70}}\!\,</math>
| <math>-\sqrt{\frac{1}{10}}\!\,</math>
| <math>\sqrt{\frac{2}{7}}\!\,</math>
| <math>-\sqrt{\frac{2}{5}}\!\,</math>
| <math>\sqrt{\frac{1}{5}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=5/2, j<sub>2</sub>=1/2===
 
{|
|----- align="center"
| m=3 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3'''
|----- align="center"
| '''5/2,&nbsp;1/2''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3''' || '''2'''
|----- align="center"
| '''5/2,&nbsp;-1/2''' || <math>\sqrt{\frac{1}{6}}\!\,</math>
| <math>\sqrt{\frac{5}{6}}\!\,</math>
|----- align="center"
| '''3/2,&nbsp;1/2''' || <math>\sqrt{\frac{5}{6}}\!\,</math>
| <math>-\sqrt{\frac{1}{6}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3''' || '''2'''
|----- align="center"
| '''3/2,&nbsp;-1/2''' || <math>\sqrt{\frac{1}{3}}\!\,</math>
| <math>\sqrt{\frac{2}{3}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;1/2''' || <math>\sqrt{\frac{2}{3}}\!\,</math>
| <math>-\sqrt{\frac{1}{3}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=0 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''3''' || '''2'''
|----- align="center"
| '''1/2,&nbsp;-1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=5/2, j<sub>2</sub>=1===
 
{|
|----- align="center"
| m=7/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''7/2'''
|----- align="center"
| '''5/2,&nbsp;1''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=5/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''7/2''' || '''5/2'''
|----- align="center"
| '''5/2,&nbsp;0''' || <math>\sqrt{\frac{2}{7}}\!\,</math>
| <math>\sqrt{\frac{5}{7}}\!\,</math>
|----- align="center"
| '''3/2,&nbsp;1''' || <math>\sqrt{\frac{5}{7}}\!\,</math>
| <math>-\sqrt{\frac{2}{7}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=3/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''7/2''' || '''5/2''' || '''3/2'''
|----- align="center"
| '''5/2,&nbsp;-1'''
| <math>\sqrt{\frac{1}{21}}\!\,</math>
| <math>\sqrt{\frac{2}{7}}\!\,</math> || <math>\sqrt{\frac{2}{3}}\!\,</math>
|----- align="center"
| '''3/2,&nbsp;0'''
| <math>\sqrt{\frac{10}{21}}\!\,</math>
| <math>\sqrt{\frac{9}{35}}\!\,</math>
| <math>-\sqrt{\frac{4}{15}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;1'''
| <math>\sqrt{\frac{10}{21}}\!\,</math>
| <math>-\sqrt{\frac{16}{35}}\!\,</math>
| <math>\sqrt{\frac{1}{15}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''7/2''' || '''5/2''' || '''3/2'''
|----- align="center"
| '''3/2,&nbsp;-1''' || <math>\sqrt{\frac{1}{7}}\!\,</math>
| <math>\sqrt{\frac{16}{35}}\!\,</math>
| <math>\sqrt{\frac{2}{5}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;0''' || <math>\sqrt{\frac{4}{7}}\!\,</math>
| <math>\sqrt{\frac{1}{35}}\!\,</math>
| <math>-\sqrt{\frac{2}{5}}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;1''' || <math>\sqrt{\frac{2}{7}}\!\,</math>
| <math>-\sqrt{\frac{18}{35}}\!\,</math>
| <math>\sqrt{\frac{1}{5}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=5/2, j<sub>2</sub>=3/2===
 
{|
|----- align="center"
| m=4 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''4'''
|----- align="center"
| '''5/2,&nbsp;3/2''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=3 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''4''' || '''3'''
|----- align="center"
| '''5/2,&nbsp;1/2''' || <math>\sqrt{\frac{3}{8}}\!\,</math>
| <math>\sqrt{\frac{5}{8}}\!\,</math>
|----- align="center"
| '''3/2,&nbsp;3/2''' || <math>\sqrt{\frac{5}{8}}\!\,</math>
| <math>-\sqrt{\frac{3}{8}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''4''' || '''3''' || '''2'''
|----- align="center"
| '''5/2,&nbsp;-1/2'''
| <math>\sqrt{\frac{3}{28}}\!\,</math>
| <math>\sqrt{\frac{5}{12}}\!\,</math>
| <math>\sqrt{\frac{10}{21}}\!\,</math>
|----- align="center"
| '''3/2,&nbsp;1/2'''
| <math>\sqrt{\frac{15}{28}}\!\,</math>
| <math>\sqrt{\frac{1}{12}}\!\,</math>
| <math>-\sqrt{\frac{8}{21}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;3/2'''
| <math>\sqrt{\frac{5}{14}}\!\,</math>
| <math>-\sqrt{\frac{1}{2}}\!\,</math>
| <math>\sqrt{\frac{1}{7}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''4''' || '''3''' || '''2''' || '''1'''
|----- align="center"
| '''5/2,&nbsp;-3/2'''
| <math>\sqrt{\frac{1}{56}}\!\,</math>
| <math>\sqrt{\frac{1}{8}}\!\,</math>
| <math>\sqrt{\frac{5}{14}}\!\,</math>
| <math>\sqrt{\frac{1}{2}}\!\,</math>
|----- align="center"
| '''3/2,&nbsp;-1/2'''
| <math>\sqrt{\frac{15}{56}}\!\,</math>
| <math>\sqrt{\frac{49}{120}}\!\,</math>
| <math>\sqrt{\frac{1}{42}}\!\,</math>
| <math>-\sqrt{\frac{3}{10}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;1/2'''
| <math>\sqrt{\frac{15}{28}}\!\,</math>
| <math>-\sqrt{\frac{1}{60}}\!\,</math>
| <math>-\sqrt{\frac{25}{84}}\!\,</math>
| <math>\sqrt{\frac{3}{20}}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;3/2'''
| <math>\sqrt{\frac{5}{28}}\!\,</math>
| <math>-\sqrt{\frac{9}{20}}\!\,</math>
| <math>\sqrt{\frac{9}{28}}\!\,</math>
| <math>-\sqrt{\frac{1}{20}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=0 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''4''' || '''3''' || '''2''' || '''1'''
|----- align="center"
| '''3/2,&nbsp;-3/2'''
| <math>\sqrt{\frac{1}{14}}\!\,</math>
| <math>\sqrt{\frac{3}{10}}\!\,</math>
| <math>\sqrt{\frac{3}{7}}\!\,</math> || <math>\sqrt{\frac{1}{5}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;-1/2''' || <math>\sqrt{\frac{3}{7}}\!\,</math>
| <math>\sqrt{\frac{1}{5}}\!\,</math>
| <math>-\sqrt{\frac{1}{14}}\!\,</math>
| <math>-\sqrt{\frac{3}{10}}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;1/2''' || <math>\sqrt{\frac{3}{7}}\!\,</math>
| <math>-\sqrt{\frac{1}{5}}\!\,</math>
| <math>-\sqrt{\frac{1}{14}}\!\,</math>
| <math>\sqrt{\frac{3}{10}}\!\,</math>
|----- align="center"
| '''-3/2,&nbsp;3/2'''
| <math>\sqrt{\frac{1}{14}}\!\,</math>
| <math>-\sqrt{\frac{3}{10}}\!\,</math>
| <math>\sqrt{\frac{3}{7}}\!\,</math>
| <math>-\sqrt{\frac{1}{5}}\!\,</math>
|}
|}
 
===j<sub>1</sub>=5/2, j<sub>2</sub>=2===
 
{|
|----- align="center"
| m=9/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''9/2'''
|----- align="center"
| '''5/2,&nbsp;2''' || <math>1\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=7/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''9/2''' || '''7/2'''
|----- align="center"
| '''5/2,&nbsp;1''' || <math>\frac{2}{3}\!\,</math>
| <math>\sqrt{\frac{5}{9}}\!\,</math>
|----- align="center"
| '''3/2,&nbsp;2''' || <math>\sqrt{\frac{5}{9}}\!\,</math>
| <math>-\frac{2}{3}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=5/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''9/2''' || '''7/2''' || '''5/2'''
|----- align="center"
| '''5/2,&nbsp;0''' || <math>\sqrt{\frac{1}{6}}\!\,</math>
| <math>\sqrt{\frac{10}{21}}\!\,</math>
| <math>\sqrt{\frac{5}{14}}\!\,</math>
|----- align="center"
| '''3/2,&nbsp;1''' || <math>\sqrt{\frac{5}{9}}\!\,</math>
| <math>\sqrt{\frac{1}{63}}\!\,</math>
| <math>-\sqrt{\frac{3}{7}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;2'''
| <math>\sqrt{\frac{5}{18}}\!\,</math>
| <math>-\sqrt{\frac{32}{63}}\!\,</math>
| <math>\sqrt{\frac{3}{14}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=3/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''9/2''' || '''7/2''' || '''5/2''' || '''3/2'''
|----- align="center"
| '''5/2,&nbsp;-1'''
| <math>\sqrt{\frac{1}{21}}\!\,</math>
| <math>\sqrt{\frac{5}{21}}\!\,</math>
| <math>\sqrt{\frac{3}{7}}\!\,</math> || <math>\sqrt{\frac{2}{7}}\!\,</math>
|----- align="center"
| '''3/2,&nbsp;0'''
| <math>\sqrt{\frac{5}{14}}\!\,</math>
| <math>\sqrt{\frac{2}{7}}\!\,</math>
| <math>-\sqrt{\frac{1}{70}}\!\,</math>
| <math>-\sqrt{\frac{12}{35}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;1'''
| <math>\sqrt{\frac{10}{21}}\!\,</math>
| <math>-\sqrt{\frac{2}{21}}\!\,</math>
| <math>-\sqrt{\frac{6}{35}}\!\,</math>
| <math>\sqrt{\frac{9}{35}}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;2'''
| <math>\sqrt{\frac{5}{42}}\!\,</math>
| <math>-\sqrt{\frac{8}{21}}\!\,</math>
| <math>\sqrt{\frac{27}{70}}\!\,</math>
| <math>-\sqrt{\frac{4}{35}}\!\,</math>
|}
|}
 
{|
|----- align="center"
| m=1/2 || j=
|-----
| <br /><br /><br />m<sub>1</sub>,&nbsp;m<sub>2</sub>=
|
{| border="1"
|-----
|
|| '''9/2''' || '''7/2''' || '''5/2''' || '''3/2'''
| '''1/2'''
|----- align="center"
| '''5/2,&nbsp;-2'''
| <math>\sqrt{\frac{1}{126}}\!\,</math>
| <math>\sqrt{\frac{4}{63}}\!\,</math>
| <math>\sqrt{\frac{3}{14}}\!\,</math>
| <math>\sqrt{\frac{8}{21}}\!\,</math>
| <math>\sqrt{\frac{1}{3}}\!\,</math>
|----- align="center"
| '''3/2,&nbsp;-1'''
| <math>\sqrt{\frac{10}{63}}\!\,</math>
| <math>\sqrt{\frac{121}{315}}\!\,</math>
| <math>\sqrt{\frac{6}{35}}\!\,</math>
| <math>-\sqrt{\frac{2}{105}}\!\,</math>
| <math>-\sqrt{\frac{4}{15}}\!\,</math>
|----- align="center"
| '''1/2,&nbsp;0'''
| <math>\sqrt{\frac{10}{21}}\!\,</math>
| <math>\sqrt{\frac{4}{105}}\!\,</math>
| <math>-\sqrt{\frac{8}{35}}\!\,</math>
| <math>-\sqrt{\frac{2}{35}}\!\,</math>
| <math>\sqrt{\frac{1}{5}}\!\,</math>
|----- align="center"
| '''-1/2,&nbsp;1'''
| <math>\sqrt{\frac{20}{63}}\!\,</math>
| <math>-\sqrt{\frac{14}{45}}\!\,</math>
| <math>0\!\,</math>
| <math>\sqrt{\frac{5}{21}}\!\,</math>
| <math>-\sqrt{\frac{2}{15}}\!\,</math>
|----- align="center"
| '''-3/2,&nbsp;2'''
| <math>\sqrt{\frac{5}{126}}\!\,</math>
| <math>-\sqrt{\frac{64}{315}}\!\,</math>
| <math>\sqrt{\frac{27}{70}}\!\,</math>
| <math>-\sqrt{\frac{32}{105}}\!\,</math>
| <math>\sqrt{\frac{1}{15}}\!\,</math>
|}
|}
 
==SU(N) Clebsch-Gordan coefficients==
 
Algorithms to produce Clebsch-Gordan coefficients for higher values of <math>j_1</math> and <math>j_2</math>, or for the su(N) algebra instead of su(2), are known.<ref>{{cite journal |last=Alex |first=A. |coauthors=M. Kalus, A. Huckleberry, and J. von Delft |title=A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients |journal=J. Math. Phys. |volume=82 |date=February 2011 |pages=023507 |doi= 10.1063/1.3521562 |url=http://link.aip.org/link/doi/10.1063/1.3521562 |accessdate=2011-04-13 |bibcode=2011JMP....52b3507A|arxiv = 1009.0437 }}</ref>
A [http://homepages.physik.uni-muenchen.de/~vondelft/Papers/ClebschGordan/ web interface for tabulating SU(N) Clebsch-Gordan coefficients] is readily available.
 
==References==
<references/>
 
==External links==
* Online, [[Java]]-based [http://personal.ph.surrey.ac.uk/~phs3ps/cgjava.html Clebsch-Gordan Coefficient Calculator] by Paul Stevenson
* [http://functions.wolfram.com/HypergeometricFunctions/ClebschGordan/06/01/ Other formulae] for Clebsch-Gordan coefficients.
*  [http://homepages.physik.uni-muenchen.de/~vondelft/Papers/ClebschGordan/ Web interface for tabulating SU(N) Clebsch-Gordan coefficients]
 
{{DEFAULTSORT:Table of Clebsch-Gordan coefficients}}
[[Category:Representation theory of Lie groups]]
[[Category:Mathematical tables|Clebsch-Gordan coefficients]]

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