Racah W-coefficient

Racah's W-coefficients were introduced by Giulio Racah in 1942. [G. Racah, "Theory of Complex Spectra II", Physical Review, vol. 62, pp. 438-462 (1942).] These coefficients have a purely mathematical definition. In physics theyare used in calculations involving the quantum mechanicaldescription of angular momentum, for example in atomic theory.

The coefficients appear when there are three sources of angular momentumin the problem. For example, consider an atom with one electron in an
s orbital and one electron in a p orbital.Each electron has electron spin angular momentum and in additionthe 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" couplingas explained in the article on angular momentum coupling.The transformation between the wave functions that correspondto these two coulings involves a Racah W-coefficient.

Apart from a phase factor, Racah's W-coefficients are equal toWigner's 6-j symbols, so any equationinvolving Racah's W-coefficients may be rewritten using6-j symbols. This is often advantageous because thesymmetry properties of 6-j symbols are easier toremember.

Racah coefficients are related to recoupling coefficients by:$W(j\_1j\_2Jj\_3;J\_\{12\}J\_\{23\})\; equiv\; [(2J\_\{12\}+1)(2J\_\{23\}+1)]\; ^\{-frac\{1\}\{2\; langle\; (j\_1,\; (j\_2j\_3)J\_\{23\})\; J\; |\; ((j\_1j\_2)J\_\{12\},j\_3)J\; angle.$Recoupling coefficients are elements of a unitary transformationand their definition is given in the next section.Racah coefficients have more convenient symmetry properties thanthe recoupling coefficients (but less convenient than the 6-j symbols).

Recoupling coefficients

Coupling of two angular momenta $mathbf\{j\}\_1$ and $mathbf\{j\}\_2$is the construction of simultaneous eigenfunctions of $mathbf\{J\}^2$and $J\_z$, where $mathbf\{J\}=mathbf\{j\}\_1+mathbf\{j\}\_2$,as explained in the article on Clebsch-Gordan coefficients. The result is:$$
(j_1j_2)JM angle = sum_{m_1=-j_1}^{j_1} sum_{m_2=-j_2}^{j_2}
j_1m_1 angle |j_2m_2 angle langle j_1m_1j_2m_2|JM angle,where $J=|j\_1-j\_2|,ldots,j\_1+j\_2$ and $M=-J,ldots,J$.

Coupling of three angular momenta $mathbf\{j\}\_1$, $mathbf\{j\}\_2$, and $mathbf\{j\}\_3$,may be done by first coupling $mathbf\{j\}\_1$ and $mathbf\{j\}\_2$ to $mathbf\{J\}\_\{12\}$and next coupling $mathbf\{J\}\_\{12\}$ and $mathbf\{j\}\_3$ to total angular momentum$mathbf\{J\}$::$$
((j_1j_2)J_{12}j_3)JM angle = sum_{M_{12}=-J_{12^{J_{12 sum_{m_3=-j_3}^{j_3}
(j_1j_2)J_{12}M_{12} angle |j_3m_3 angle langle J_{12}M_{12}j_3m_3|JM angle

Alternatively, one may first couple $mathbf\{j\}\_2$ and $mathbf\{j\}\_3$to $mathbf\{J\}\_\{23\}$ and next couple $mathbf\{j\}\_1$ and $mathbf\{J\}\_\{23\}$to $mathbf\{J\}$::$$
(j_1,(j_2j_3)J_{23})JM angle = sum_{m_1=-j_1}^{j_1} sum_{M_{23}=-J_{23^{J_{23
j_1m_1 angle |(j_2j_3)J_{23}M_{23} angle langle j_1m_1J_{23}M_{23}|JM angle

Both coupling schemes result in complete orthonormal bases for the$(2j\_1+1)(2j\_2+1)(2j\_3+1)$ dimensional space spanned by:$$
j_1 m_1 angle |j_2 m_2 angle |j_3 m_3 angle, ;; m_1=-j_1,ldots,j_1;;; m_2=-j_2,ldots,j_2;;; m_3=-j_3,ldots,j_3.Hence, the two total angular momentum bases are related by a unitary transformation. The matrix elementsof this unitary transformation are given by a scalar product and are known as recouplingcoefficients. The coefficients are independent of $M$ and so we have:$$
((j_1j_2)J_{12}j_3)JM angle = sum_{J_{23 |(j_1,(j_2j_3)J_{23})JM angle langle (j_1,(j_2j_3)J_{23})J |((j_1j_2)J_{12}j_3)J angle.The independence of $M$ follows readily by writing this equation for $M=J$and applying the lowering operator$J\_-$ to both sides of the equation.

Relation to Wigner's 6-j symbol

Racah's W-coefficients are related to Wigner's 6-j symbols, which have even moreconvenient symmetry properties:

ee also

* Clebsch-Gordan coefficient
* 3-jm symbol
* 6-j symbol

Cited reference

Further references

* cite book |last= Edmonds |first= A. R. |title= Angular Momentum in Quantum Mechanics |year= 1957
publisher= Princeton University Press |location= Princeton, New Jersey |isbn= 0-691-07912-9

* cite book |last= Condon |first= Edward U. |coauthors= Shortley, G. H. |title= The Theory of Atomic Spectra |year= 1970
publisher= Cambridge University Press |location= Cambridge |isbn= 0-521-09209-4 |chapter= Chapter 3

* cite book |last= Messiah |first= Albert |title= Quantum Mechanics (Volume II) |year= 1981 | edition= 12th edition
publisher= North Holland Publishing |location= New York |isbn= 0-7204-0045-7

* cite book |last= Brink |first= D. M. |coauthors= Satchler, G. R. |title= Angular Momentum
year= 1993 |edition= 3rd edition |publisher= Clarendon Press |location= Oxford |isbn= 0-19-851759-9 |chapter= Chapter 2

* cite book |last= Zare |first= Richard N. |title= Angular Momentum |year=1988
publisher= John Wiley |location= New York |isbn= 0-471-85892-7 |chapter= Chapter 2

* cite book |last= Biedenharn |first= L. C. |coauthors= Louck, J. D. |title= Angular Momentum in Quantum Physics
year= 1981 |publisher= Addison-Wesley |location= Reading, Massachusetts |isbn= 0201135078