词条 | Racah W-coefficient |
释义 |
Racah's W-coefficients were introduced by Giulio Racah in 1942.[1] These coefficients have a purely mathematical definition. In physics they are used in calculations involving the quantum mechanical description of angular momentum, for example in atomic theory. The coefficients appear when there are three sources of angular momentum in 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 addition 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. Apart from a phase factor, Racah's W-coefficients are equal to Wigner's 6-j symbols, 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. Racah coefficients are related to recoupling coefficients by 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).[2] Recoupling coefficientsCoupling of two angular momenta and is the construction of simultaneous eigenfunctions of and , where , as explained in the article on Clebsch–Gordan coefficients. The result is where and . Coupling of three angular momenta , , and , may be done by first coupling and to and next coupling and to total angular momentum : Alternatively, one may first couple and to and next couple and to : Both coupling schemes result in complete orthonormal bases for the dimensional space spanned by 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 and so we have The independence of follows readily by writing this equation for and applying the lowering operator to both sides of the equation. AlgebraLet be the usual triangular factor, then the Racah coefficient is a product of four of these by a sum over factorials, where and The sum over is finite over the range[3] Relation to Wigner's 6-j symbolRacah's W-coefficients are related to Wigner's 6-j symbols, which have even more convenient symmetry properties Cf.[4] or See also
Notes1. ^{{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 }} 2. ^Rose, M. E. (1957). Elementary theory of angular momentum (Dover). 3. ^Cowan, R D (1981). The theory of atomic structure and spectra (Univ of California Press), p. 148. 4. ^Brink, D M & Satchler, G R (1968). Angular Momentum (Oxford University Press) 3 ed., p. 142. [https://archive.org/details/AngularMomentum online] Further reading
|publisher= Princeton University Press |location= Princeton, New Jersey |isbn= 0-691-07912-9}}
|publisher= Cambridge University Press |location= Cambridge |isbn= 0-521-09209-4 |chapter= Chapter 3}}
|publisher= North Holland Publishing |location= New York |isbn= 0-7204-0045-7}}
|first=Masachiyo |last1=Sato |journal=Progr. Theor. Physics |volume=13 |issue=4 |year=1955 |pages=405–414 |title=General formula of the Racah coefficient |doi=10.1143/PTP.13.405 |bibcode=1955PThPh..13..405S }}
|year= 1993 |edition= 3rd |publisher= Clarendon Press |location= Oxford |isbn= 0-19-851759-9 |chapter= Chapter 2 }}
|publisher= John Wiley |location= New York |isbn= 0-471-85892-7 |chapter= Chapter 2}}
|year= 1981 |publisher= Addison-Wesley |location= Reading, Massachusetts |isbn= 0-201-13507-8 }} External links
2 : Rotational symmetry|Representation theory of Lie groups |
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