Mar 14, 1999
PHY 742 -- Notes on Clebsch-GordanPHY 742 - Notes on Clebsch-Gordan coefficients
Reference: Abramowitz & Stegum - pg. 1006
Original formula:
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< j1 j2 m1 m2|j1 j2 j m > = dm,m1+m2 |
æ
Ö
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(j1+j2-j)!(j1-j2+j)!(-j1+j2+j)!(2j+1)
(j1+j2+j+1)!
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× |
å
k
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(-1)k |
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Ö(j1+m1)!(j1-m1)!(j2+m2)!(j2-m2)!(j+m)!(j-m)!
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k!(j1+j2-j-k)!(j1-m1-k)!(j2+m2-k)!(j-j2+m1+k)!(j-j1-m2+k)!
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Working formula:
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< j1 j2 m1 m2|j1 j2 j m > = dm,m1+m2 |
æ
Ö
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(j1-j2+j)!(-j1+j2+j)!(2j+1)
(j1+j2-j)!(j1+j2+j+1)!
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| (2) | |
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× |
æ
Ö
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(j1+m1)!(j2-m2)!(j+m)!
(j1-m1)!(j2+m2)!(j-m)!
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× |
å
k
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(-1)k
k!
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(j1+j2-j)!
(j1+j2-j-k)!
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(j1-m1)!
(j1-m1-k)!
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(j2+m2)!
(j2+m2-k)!
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(j-m)!
(j-j2+m1+k)!(j-j1-m2+k)!
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Additional reference for Clebsch-Gordan coefficients: Complement
B_X and C_X of Cohen-Tannoudji's text (Volume #2).
A related quantity is the Gaunt coefficient:
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GL Ml1m1 l2m2 º |
æ
Ö
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4 p
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ó
õ
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dWY*l1m1( |
^
r
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)Y*LM( |
^
r
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)Yl2m2( |
^
r
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). |
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As discussed in Cohen-Tannoudji's text, the Gaunt coefficent can be expressed
in terms of a product of two Clebsch-Gordan coefficients and hence
can also be calculated analytically.
File translated from TEX by TTH, version 1.92.
On 14 Mar 1999, 17:07.