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| '''Racah's W-coefficients''' were introduced by [[Giulio Racah]] in 1942.<ref>{{Cite journal |first=G. |last=Racah |title=Theory of Complex Spectra II |journal=[[Physical Review]] |volume=62 |issue=9–10 |pages=438–462 |year=1942 |doi=10.1103/PhysRev.62.438 |bibcode = 1942PhRv...62..438R }}</ref> These coefficients have a purely mathematical definition. In physics they are used in calculations involving the [[quantum mechanics|quantum mechanical]] description of [[angular momentum]], for example in [[atomic theory]].
| | The writer is known by the title of Numbers Lint. Her family members lives in Minnesota. My working day job is a librarian. Doing ceramics is what my family members and I appreciate.<br><br>Have a look at my weblog ... [http://Essexsinglesnights.co.uk/groups/cures-for-any-candida-tips-to-use-now/ at home std test] |
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| The coefficients appear when there are three sources of angular momentum in the problem. For example, consider an atom with one electron in an [[atomic orbital|s orbital]] and one electron in a [[atomic orbital|p orbital]]. Each electron has [[electron spin]] angular momentum and in addition
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| the p orbital has orbital angular momentum (an s orbital has zero orbital angular momentum). The atom may be described by ''LS'' coupling or by ''jj'' coupling as explained in the article on [[angular momentum coupling]]. The transformation between the wave functions that correspond to these two couplings involves a Racah W-coefficient.
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| Apart from a phase factor, Racah's W-coefficients are equal to Wigner's [[6-j symbol]]s, so any equation involving Racah's W-coefficients may be rewritten using 6-j symbols. This is often advantageous because the symmetry properties of 6-j symbols are easier to remember.
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| Racah coefficients are related to recoupling coefficients by
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| :<math>
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| W(j_1j_2Jj_3;J_{12}J_{23}) \equiv [(2J_{12}+1)(2J_{23}+1)]^{-\frac{1}{2}}
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| \langle (j_1, (j_2j_3)J_{23}) J | ((j_1j_2)J_{12},j_3)J \rangle.
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| </math>
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| Recoupling coefficients are elements of a [[unitary transformation]] and their definition is given in the next section. Racah coefficients have more convenient symmetry properties than the recoupling coefficients (but less convenient than the 6-j symbols).
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| ==Recoupling coefficients==
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| Coupling of two angular momenta <math>\mathbf{j}_1</math> and <math>\mathbf{j}_2</math> is the construction of simultaneous eigenfunctions of <math>\mathbf{J}^2</math> and <math>J_z</math>, where <math>\mathbf{J}=\mathbf{j}_1+\mathbf{j}_2</math>, as explained in the article on [[Clebsch-Gordan coefficients]]. The result is
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| :<math>
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| |(j_1j_2)JM\rangle = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2}
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| |j_1m_1\rangle |j_2m_2\rangle \langle j_1m_1j_2m_2|JM\rangle,
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| </math>
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| where <math>J=|j_1-j_2|,\ldots,j_1+j_2</math> and <math>M=-J,\ldots,J</math>.
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| Coupling of three angular momenta <math>\mathbf{j}_1</math>, <math>\mathbf{j}_2</math>, and <math>\mathbf{j}_3</math>, may be done by first coupling <math>\mathbf{j}_1</math> and <math>\mathbf{j}_2</math> to <math>\mathbf{J}_{12}</math> and next coupling <math>\mathbf{J}_{12}</math> and <math>\mathbf{j}_3</math> to total angular momentum <math>\mathbf{J}</math>:
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| :<math>
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| |((j_1j_2)J_{12}j_3)JM\rangle = \sum_{M_{12}=-J_{12}}^{J_{12}} \sum_{m_3=-j_3}^{j_3}
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| |(j_1j_2)J_{12}M_{12}\rangle |j_3m_3\rangle \langle J_{12}M_{12}j_3m_3|JM\rangle
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| </math>
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| Alternatively, one may first couple <math>\mathbf{j}_2</math> and <math>\mathbf{j}_3</math> to <math>\mathbf{J}_{23}</math> and next couple <math>\mathbf{j}_1</math> and <math>\mathbf{J}_{23}</math> to <math>\mathbf{J}</math>:
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| :<math>
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| |(j_1,(j_2j_3)J_{23})JM \rangle = \sum_{m_1=-j_1}^{j_1} \sum_{M_{23}=-J_{23}}^{J_{23}}
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| |j_1m_1\rangle |(j_2j_3)J_{23}M_{23}\rangle \langle j_1m_1J_{23}M_{23}|JM\rangle
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| </math>
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| Both coupling schemes result in complete orthonormal bases for the <math>(2j_1+1)(2j_2+1)(2j_3+1)</math> dimensional space spanned by
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| :<math>
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| |j_1 m_1\rangle |j_2 m_2\rangle |j_3 m_3\rangle, \;\; m_1=-j_1,\ldots,j_1;\;\; m_2=-j_2,\ldots,j_2;\;\; m_3=-j_3,\ldots,j_3.
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| </math>
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| Hence, the two total angular momentum bases are related by a unitary transformation. The matrix elements of this unitary transformation are given by a [[scalar product]] and are known as recoupling coefficients. The coefficients are independent of <math>M</math> and so we have
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| :<math>
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| |((j_1j_2)J_{12}j_3)JM\rangle = \sum_{J_{23}} |(j_1,(j_2j_3)J_{23})JM \rangle
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| \langle (j_1,(j_2j_3)J_{23})J |((j_1j_2)J_{12}j_3)J\rangle.
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| </math>
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| The independence of <math>M</math> follows readily by writing this equation for <math>M=J</math> and applying the [[Clebsch-Gordan coefficient#Angular momentum operators|lowering operator]] <math>J_-</math> to both sides of the equation.
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| ==Algebra==
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| Let
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| :<math>\Delta(a,b,c)=[(a+b-c)!(a-b+c)!(-a+b+c)!/(a+b+c+1)!]^{1/2}</math>
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| be the usual triangular factor, then the Racah coefficient is a product
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| of four of these by a sum over factorials,
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| :<math>W(abcd;ef)=\Delta(a,b,e)\Delta(c,d,e)\Delta(a,c,f)\Delta(b,d,f)w(abcd;ef)
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| </math>
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| where
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| :<math>w(abcd;ef)\equiv
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| \sum_z\frac{(-1)^{z+\beta_1}(z+1)!}{(z-\alpha_1)!(z-\alpha_2)!(z-\alpha_3)!
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| (z-\alpha_4)!(\beta_1-z)!(\beta_2-z)!(\beta_3-z)!}</math>
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| and
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| :<math>\alpha_1=a+b+e;\quad \beta_1=a+b+c+d;</math> | |
| :<math>\alpha_2=c+d+e;\quad \beta_2=a+d+e+f;</math>
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| :<math>\alpha_3=a+c+f;\quad \beta_3=b+c+e+f;</math>
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| :<math>\alpha_4=b+d+f.</math>
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| The sum over <math>z</math> is finite over the range<ref>The Theory of Atomic Structure and Spectra, Cowan, p. 148</ref>
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| :<math> \max(\alpha_1,\alpha_2,\alpha_3,\alpha_4) \le z \le \min(\beta_1,\beta_2,\beta_3). </math>
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| ==Relation to Wigner's 6-j symbol==
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| Racah's W-coefficients are related to Wigner's [[6-j symbol]]s, which have even more convenient symmetry properties
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| :<math>
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| W(abcd;ef)(-1)^{a+b+c+d}=
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| \begin{Bmatrix}
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| a&b&e\\
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| d&c&f
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| \end{Bmatrix}.
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| </math>
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| See <ref>Angular momentum 3<math>^{rd}</math> ed. by D.M. Brink and G.R. Satchler Page 142 </ref> or
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| :<math>
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| W(j_1j_2Jj_3;J_{12}J_{23}) = (-1)^{j_1+j_2+j_3+J}
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| \begin{Bmatrix}
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| j_1 & j_2 & J_{12}\\
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| j_3 & J & J_{23}
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| \end{Bmatrix}.
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| </math>
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| ==See also==
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| * [[Clebsch–Gordan coefficients]]
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| * [[3-jm symbol]]
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| * [[6-j symbol]]
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| ==Notes==
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| {{Reflist}}
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| ==Further reading==
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| * {{cite book |last= Edmonds |first= A. R. |title= Angular Momentum in Quantum Mechanics |year= 1957
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| |publisher= [[Princeton University Press]] |location= Princeton, New Jersey |isbn= 0-691-07912-9}}
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| * {{cite book |last= Condon |first= Edward U. |coauthors= Shortley, G. H. |title= The Theory of Atomic Spectra |year= 1970
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| |publisher= [[Cambridge University Press]] |location= Cambridge |isbn= 0-521-09209-4 |chapter= Chapter 3}}
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| * {{cite book |last= Messiah |first= Albert |title= Quantum Mechanics (Volume II) |year= 1981 | edition= 12th edition
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| |publisher= [[Elsevier|North Holland Publishing]] |location= New York |isbn= 0-7204-0045-7}}
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| *{{cite journal
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| |first=Masachiyo
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| |last1=Sato
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| |journal=Progr. Theor. Physics
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| |volume=13
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| |issue=4
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| |year=1955
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| |pages=405–414
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| |title=General formula of the Racah coefficient
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| |doi=10.1143/PTP.13.405
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| |bibcode=1955PThPh..13..405S
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| }}
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| * {{cite book |last= Brink |first= D. M. |coauthors= Satchler, G. R. |title= Angular Momentum
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| |year= 1993 |edition= 3rd edition |publisher= [[Clarendon Press]] |location= Oxford |isbn= 0-19-851759-9 |chapter= Chapter 2 }}
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| * {{cite book |last= Zare |first= Richard N. |title= Angular Momentum |year=1988
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| |publisher= [[John Wiley & Sons|John Wiley]] |location= New York |isbn= 0-471-85892-7 |chapter= Chapter 2}}
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| * {{cite book |last= Biedenharn |first= L. C. |coauthors= Louck, J. D. |title= Angular Momentum in Quantum Physics
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| |year= 1981 |publisher= [[Addison-Wesley]] |location= Reading, Massachusetts |isbn= 0-201-13507-8 }}
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| ==External links==
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| * {{springer|title=Racah-Wigner coefficients|id=p/r110010}}
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| [[Category:Rotational symmetry]]
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| [[Category:Representation theory of Lie groups]]
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The writer is known by the title of Numbers Lint. Her family members lives in Minnesota. My working day job is a librarian. Doing ceramics is what my family members and I appreciate.
Have a look at my weblog ... at home std test