Racah W-coefficient explained

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 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 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

W(j_1j_2Jj_3;J₁₂J₂₃)\equiv

	\langle(j_1,(j_2j_3)J₂₃)J\|((j_1j_2)J₁₂,j_3)J\rangle
	\sqrt{(2J₁₂+1)(2J₂₃+1)

}.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 coefficients

Coupling of two angular momenta

j₁

and

j₂

is the construction of simultaneous eigenfunctions of

J²

and

J_z

, where

J=j_1+j₂

, as explained in the article on Clebsch–Gordan coefficients. The result is

|(j_1j_2)JM\rangle=

	j₁
\sum
	m_1=-j₁

	j₂
\sum
	m_2=-j₂

|j_1m_1\rangle|j_2m_2\rangle\langlej_1m_1j_2m_2|JM\rangle,

where

J=|j_1-j_2|,\ldots,j_1+j₂

and

M=-J,\ldots,J

Coupling of three angular momenta

j₁

j₂

, and

j₃

, may be done by first coupling

j₁

and

j₂

J₁₂

and next coupling

J₁₂

and

j₃

to total angular momentum

|((j_1j_2)J₁₂j_3)JM\rangle=

	J₁₂
\sum
	M₁₂=-J₁₂

	j₃
\sum
	m_3=-j₃

|(j_1j_2)J₁₂M₁₂\rangle|j_3m_3\rangle\langleJ₁₂M₁₂j_3m_3|JM\rangle

Alternatively, one may first couple

j₂

and

j₃

J₂₃

and next couple

j₁

and

J₂₃

|(j_1,(j_2j_3)J₂₃)JM\rangle=

	j₁
\sum
	m_1=-j₁

	J₂₃
\sum
	M₂₃=-J₂₃

|j_1m_1\rangle|(j_2j_3)J₂₃M₂₃\rangle\langlej_1m_1J₂₃M₂₃|JM\rangle

Both coupling schemes result in complete orthonormal bases for the

(2j_1+1)(2j_2+1)(2j₃₊₁₎

dimensional space spanned by

|j₁m_1\rangle|j₂m_2\rangle|j₃m_3\rangle, 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 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

|((j_1j_2)J₁₂j_3)JM\rangle=

\sum
	J₂₃

|(j_1,(j_2j_3)J₂₃)JM\rangle \langle(j_1,(j_2j_3)J₂₃)J|((j_1j_2)J₁₂j_3)J\rangle.

The independence of

follows readily by writing this equation for

M=J

and applying the lowering operator

J_-

to both sides of the equation.The definition of Racah W-coefficients lets us write this final expression as

|((j_1j_2)J₁₂j_3)JM\rangle=

\sum
	J₂₃

|(j_1,(j_2j_3)J₂₃)JM\rangle W(j_1j_2Jj_3;J₁₂J₂₃)\sqrt{(2J₁₂+1)(2J₂₃+1)}.

Algebra

Let

\Delta(a,b,c)=[(a+b-c)!(a-b+c)!(-a+b+c)!/(a+b+c+1)!]^1/2

be the usual triangular factor, then the Racah coefficient is a productof four of these by a sum over factorials,

W(abcd;ef)=\Delta(a,b,e)\Delta(c,d,e)\Delta(a,c,f)\Delta(b,d,f)w(abcd;ef)

where

w(abcd;ef)\equiv \sum

	z+\beta₁
(-1)		(z+1)!

(z-\alpha_1)!(z-\alpha_2)!(z-\alpha_{3)!
(z-\alpha}_4)!(\beta_1-z)!(\beta_2-z)!(\beta_3-z)!

and

\alpha_1=a+b+e;\beta_1=a+b+c+d;

\alpha_2=c+d+e;\beta_2=a+d+e+f;

\alpha_3=a+c+f;\beta_3=b+c+e+f;

\alpha_4=b+d+f.

The sum over

is finite over the range^[3]

max(\alpha_1,\alpha_2,\alpha_3,\alpha₄₎\lez\lemin(\beta_1,\beta_2,\beta_3).

Relation to Wigner's 6-j symbol

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

W(abcd;ef)(-1)^a+b+c+d= \begin{Bmatrix} a&b&e\\ d&c&f \end{Bmatrix}.

Cf.^[4] or

W(j_1j_2Jj_3;J₁₂J₂₃)=

	j_1+j_2+j_3+J
(-1)

\begin{Bmatrix} j₁&j₂&J₁₂\\ j₃&J&J₂₃\end{Bmatrix}.

Notes and References

G. . Racah . Theory of Complex Spectra II . . 62 . 9–10 . 438–462 . 1942 . 10.1103/PhysRev.62.438 . 1942PhRv...62..438R .
Rose, M. E. (1957). Elementary theory of angular momentum (Dover).
Cowan, R D (1981). The theory of atomic structure and spectra (Univ of California Press), p. 148.
Brink, D M & Satchler, G R (1968). Angular Momentum (Oxford University Press) 3
^rd

ed., p. 142. online

Racah W-coefficient explained

Recoupling coefficients

Algebra

Relation to Wigner's 6-j symbol

See also

Further reading

Notes and References