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| <!--
| | Nice to meet you, my name is Figures Held though I don't really like being known as like that. In her expert life she is a payroll clerk but she's always wanted her personal business. To gather badges is what her family and her enjoy. For a whilst I've been in South Dakota and my parents reside nearby.<br><br>Here is my web page: over the counter std test - [http://www.saahbi.com/index.php?do=/profile-1730/info/ navigate to these guys] - |
| :''This article is incomplete due to technical limitations.''
| |
| -->
| |
| 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''1'''
| |
| |----- align="center"
| |
| | '''1/2, 1/2''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=0 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''1''' || '''0'''
| |
| |----- align="center"
| |
| | '''1/2, -1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| |----- align="center"
| |
| | '''-1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3/2'''
| |
| |----- align="center"
| |
| | '''1, 1/2''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=1/2 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3/2''' || '''1/2'''
| |
| |----- align="center"
| |
| | '''1, -1/2''' || <math>\sqrt{\frac{1}{3}}\!\,</math>
| |
| | <math>\sqrt{\frac{2}{3}}\!\,</math>
| |
| |----- align="center"
| |
| | '''0, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''2'''
| |
| |----- align="center"
| |
| | '''1, 1''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=1 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''2''' || '''1'''
| |
| |----- align="center"
| |
| | '''1, 0''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| |----- align="center"
| |
| | '''0, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''2''' || '''1''' || '''0'''
| |
| |----- align="center"
| |
| | '''1, -1''' || <math>\sqrt{\frac{1}{6}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{2}}\!\,</math> || <math>\sqrt{\frac{1}{3}}\!\,</math>
| |
| |----- align="center"
| |
| | '''0, 0''' || <math>\sqrt{\frac{2}{3}}\!\,</math>
| |
| | <math>0\!\,</math>
| |
| | <math>-\sqrt{\frac{1}{3}}\!\,</math>
| |
| |----- align="center"
| |
| | '''-1, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''2'''
| |
| |----- align="center"
| |
| | '''3/2, 1/2''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=1 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''2''' || '''1'''
| |
| |----- align="center"
| |
| | '''3/2, -1/2''' || <math>\frac{1}{2}\!\,</math>
| |
| | <math>\sqrt{\frac{3}{4}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''2''' || '''1'''
| |
| |----- align="center"
| |
| | '''1/2, -1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| |----- align="center"
| |
| | '''-1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''5/2'''
| |
| |----- align="center"
| |
| | '''3/2, 1''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=3/2 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''5/2''' || '''3/2'''
| |
| |----- align="center"
| |
| | '''3/2, 0''' || <math>\sqrt{\frac{2}{5}}\!\,</math>
| |
| | <math>\sqrt{\frac{3}{5}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''5/2''' || '''3/2''' || '''1/2'''
| |
| |----- align="center"
| |
| | '''3/2, -1'''
| |
| | <math>\sqrt{\frac{1}{10}}\!\,</math>
| |
| | <math>\sqrt{\frac{2}{5}}\!\,</math> || <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1/2, 0''' || <math>\sqrt{\frac{3}{5}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{15}}\!\,</math>
| |
| | <math>-\sqrt{\frac{1}{3}}\!\,</math>
| |
| |----- align="center"
| |
| | '''-1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3'''
| |
| |----- align="center"
| |
| | '''3/2, 3/2''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=2 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3''' || '''2'''
| |
| |----- align="center"
| |
| | '''3/2, 1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3''' || '''2''' || '''1'''
| |
| |----- align="center"
| |
| | '''3/2, -1/2''' || <math>\sqrt{\frac{1}{5}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| | <math>\sqrt{\frac{3}{10}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1/2, 1/2''' || <math>\sqrt{\frac{3}{5}}\!\,</math>
| |
| | <math>0\!\,</math>
| |
| | <math>-\sqrt{\frac{2}{5}}\!\,</math>
| |
| |----- align="center"
| |
| | '''-1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3''' || '''2''' || '''1''' || '''0'''
| |
| |----- align="center"
| |
| | '''3/2, -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, -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, 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, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''5/2'''
| |
| |----- align="center"
| |
| | '''2, 1/2''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=3/2 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''5/2''' || '''3/2'''
| |
| |----- align="center"
| |
| | '''2, -1/2''' || <math>\sqrt{\frac{1}{5}}\!\,</math>
| |
| | <math>\sqrt{\frac{4}{5}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''5/2''' || '''3/2'''
| |
| |----- align="center"
| |
| | '''1, -1/2''' || <math>\sqrt{\frac{2}{5}}\!\,</math>
| |
| | <math>\sqrt{\frac{3}{5}}\!\,</math>
| |
| |----- align="center"
| |
| | '''0, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3'''
| |
| |----- align="center"
| |
| | '''2, 1''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=2 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3''' || '''2'''
| |
| |----- align="center"
| |
| | '''2, 0''' || <math>\sqrt{\frac{1}{3}}\!\,</math>
| |
| | <math>\sqrt{\frac{2}{3}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3''' || '''2''' || '''1'''
| |
| |----- align="center"
| |
| | '''2, -1'''
| |
| | <math>\sqrt{\frac{1}{15}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{3}}\!\,</math> || <math>\sqrt{\frac{3}{5}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1, 0'''
| |
| | <math>\sqrt{\frac{8}{15}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{6}}\!\,</math>
| |
| | <math>-\sqrt{\frac{3}{10}}\!\,</math>
| |
| |----- align="center"
| |
| | '''0, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3''' || '''2''' || '''1'''
| |
| |----- align="center"
| |
| | '''1, -1''' || <math>\sqrt{\frac{1}{5}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| | <math>\sqrt{\frac{3}{10}}\!\,</math>
| |
| |----- align="center"
| |
| | '''0, 0''' || <math>\sqrt{\frac{3}{5}}\!\,</math>
| |
| | <math>0\!\,</math>
| |
| | <math>-\sqrt{\frac{2}{5}}\!\,</math>
| |
| |----- align="center"
| |
| | '''-1, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''7/2'''
| |
| |----- align="center"
| |
| | '''2, 3/2''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=5/2 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''7/2''' || '''5/2'''
| |
| |----- align="center"
| |
| | '''2, 1/2''' || <math>\sqrt{\frac{3}{7}}\!\,</math>
| |
| | <math>\sqrt{\frac{4}{7}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''7/2''' || '''5/2''' || '''3/2'''
| |
| |----- align="center"
| |
| | '''2, -1/2''' || <math>\sqrt{\frac{1}{7}}\!\,</math>
| |
| | <math>\sqrt{\frac{16}{35}}\!\,</math>
| |
| | <math>\sqrt{\frac{2}{5}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1, 1/2''' || <math>\sqrt{\frac{4}{7}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{35}}\!\,</math>
| |
| | <math>-\sqrt{\frac{2}{5}}\!\,</math>
| |
| |----- align="center"
| |
| | '''0, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''7/2''' || '''5/2''' || '''3/2''' || '''1/2'''
| |
| |----- align="center"
| |
| | '''2, -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, -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, 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, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''4'''
| |
| |----- align="center"
| |
| | '''2, 2''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=3 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''4''' || '''3'''
| |
| |----- align="center"
| |
| | '''2, 1''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''4''' || '''3''' || '''2'''
| |
| |----- align="center"
| |
| | '''2, 0'''
| |
| | <math>\sqrt{\frac{3}{14}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{2}}\!\,</math> || <math>\sqrt{\frac{2}{7}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1, 1''' || <math>\sqrt{\frac{4}{7}}\!\,</math>
| |
| | <math>0\!\,</math>
| |
| | <math>-\sqrt{\frac{3}{7}}\!\,</math>
| |
| |----- align="center"
| |
| | '''0, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''4''' || '''3''' || '''2''' || '''1'''
| |
| |----- align="center"
| |
| | '''2, -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, 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, 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, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''4''' || '''3''' || '''2''' || '''1'''
| |
| | '''0'''
| |
| |----- align="center"
| |
| | '''2, -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, -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, 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, 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, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3'''
| |
| |----- align="center"
| |
| | '''5/2, 1/2''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=2 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3''' || '''2'''
| |
| |----- align="center"
| |
| | '''5/2, -1/2''' || <math>\sqrt{\frac{1}{6}}\!\,</math>
| |
| | <math>\sqrt{\frac{5}{6}}\!\,</math>
| |
| |----- align="center"
| |
| | '''3/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3''' || '''2'''
| |
| |----- align="center"
| |
| | '''3/2, -1/2''' || <math>\sqrt{\frac{1}{3}}\!\,</math>
| |
| | <math>\sqrt{\frac{2}{3}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''3''' || '''2'''
| |
| |----- align="center"
| |
| | '''1/2, -1/2''' || <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{2}}\!\,</math>
| |
| |----- align="center"
| |
| | '''-1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''7/2'''
| |
| |----- align="center"
| |
| | '''5/2, 1''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=5/2 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''7/2''' || '''5/2'''
| |
| |----- align="center"
| |
| | '''5/2, 0''' || <math>\sqrt{\frac{2}{7}}\!\,</math>
| |
| | <math>\sqrt{\frac{5}{7}}\!\,</math>
| |
| |----- align="center"
| |
| | '''3/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''7/2''' || '''5/2''' || '''3/2'''
| |
| |----- align="center"
| |
| | '''5/2, -1'''
| |
| | <math>\sqrt{\frac{1}{21}}\!\,</math>
| |
| | <math>\sqrt{\frac{2}{7}}\!\,</math> || <math>\sqrt{\frac{2}{3}}\!\,</math>
| |
| |----- align="center"
| |
| | '''3/2, 0'''
| |
| | <math>\sqrt{\frac{10}{21}}\!\,</math>
| |
| | <math>\sqrt{\frac{9}{35}}\!\,</math>
| |
| | <math>-\sqrt{\frac{4}{15}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''7/2''' || '''5/2''' || '''3/2'''
| |
| |----- align="center"
| |
| | '''3/2, -1''' || <math>\sqrt{\frac{1}{7}}\!\,</math>
| |
| | <math>\sqrt{\frac{16}{35}}\!\,</math>
| |
| | <math>\sqrt{\frac{2}{5}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1/2, 0''' || <math>\sqrt{\frac{4}{7}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{35}}\!\,</math>
| |
| | <math>-\sqrt{\frac{2}{5}}\!\,</math>
| |
| |----- align="center"
| |
| | '''-1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''4'''
| |
| |----- align="center"
| |
| | '''5/2, 3/2''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=3 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''4''' || '''3'''
| |
| |----- align="center"
| |
| | '''5/2, 1/2''' || <math>\sqrt{\frac{3}{8}}\!\,</math>
| |
| | <math>\sqrt{\frac{5}{8}}\!\,</math>
| |
| |----- align="center"
| |
| | '''3/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''4''' || '''3''' || '''2'''
| |
| |----- align="center"
| |
| | '''5/2, -1/2'''
| |
| | <math>\sqrt{\frac{3}{28}}\!\,</math>
| |
| | <math>\sqrt{\frac{5}{12}}\!\,</math>
| |
| | <math>\sqrt{\frac{10}{21}}\!\,</math>
| |
| |----- align="center"
| |
| | '''3/2, 1/2'''
| |
| | <math>\sqrt{\frac{15}{28}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{12}}\!\,</math>
| |
| | <math>-\sqrt{\frac{8}{21}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''4''' || '''3''' || '''2''' || '''1'''
| |
| |----- align="center"
| |
| | '''5/2, -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, -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, 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, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''4''' || '''3''' || '''2''' || '''1'''
| |
| |----- align="center"
| |
| | '''3/2, -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, -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, 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, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''9/2'''
| |
| |----- align="center"
| |
| | '''5/2, 2''' || <math>1\!\,</math>
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |----- align="center"
| |
| | m=7/2 || j=
| |
| |-----
| |
| | <br /><br /><br />m<sub>1</sub>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''9/2''' || '''7/2'''
| |
| |----- align="center"
| |
| | '''5/2, 1''' || <math>\frac{2}{3}\!\,</math>
| |
| | <math>\sqrt{\frac{5}{9}}\!\,</math>
| |
| |----- align="center"
| |
| | '''3/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''9/2''' || '''7/2''' || '''5/2'''
| |
| |----- align="center"
| |
| | '''5/2, 0''' || <math>\sqrt{\frac{1}{6}}\!\,</math>
| |
| | <math>\sqrt{\frac{10}{21}}\!\,</math>
| |
| | <math>\sqrt{\frac{5}{14}}\!\,</math>
| |
| |----- align="center"
| |
| | '''3/2, 1''' || <math>\sqrt{\frac{5}{9}}\!\,</math>
| |
| | <math>\sqrt{\frac{1}{63}}\!\,</math>
| |
| | <math>-\sqrt{\frac{3}{7}}\!\,</math>
| |
| |----- align="center"
| |
| | '''1/2, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''9/2''' || '''7/2''' || '''5/2''' || '''3/2'''
| |
| |----- align="center"
| |
| | '''5/2, -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, 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, 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, 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>, m<sub>2</sub>=
| |
| |
| |
| {| border="1"
| |
| |-----
| |
| |
| |
| || '''9/2''' || '''7/2''' || '''5/2''' || '''3/2'''
| |
| | '''1/2'''
| |
| |----- align="center"
| |
| | '''5/2, -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, -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, 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, 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, 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]]
| |