Racah W-coefficient
Encyclopedia
Racah's W-coefficients were introduced by Giulio Racah in 1942. 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).
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.
be the usual triangular factor, then the Racah coefficient is a product
of four of these by a sum over factorials,
where
and
See or
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
description of angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
, for example in atomic theory
Atomic theory
In chemistry and physics, atomic theory is a theory of the nature of matter, which states that matter is composed of discrete units called atoms, as opposed to the obsolete notion that matter could be divided into any arbitrarily small quantity...
.
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
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
and one electron in a p orbital
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
. 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
Angular momentum coupling
In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact through spin-orbit interaction, in which case the...
. 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
Unitary transformation
In mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner product after the 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).
Recoupling coefficients
Coupling of two angular momenta and is the construction of simultaneous eigenfunctions of and , where , as explained in the article on Clebsch-Gordan coefficientsClebsch-Gordan coefficients
In physics, the Clebsch–Gordan coefficients are sets of numbers that arise in angular momentum coupling under the laws of quantum mechanics.In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum...
. 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 byHence, 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 haveThe independence of
follows readily by writing this equation for and applying the lowering operator to both sides of the equation.Algebra
Letbe the usual triangular factor, then the Racah coefficient is a product
of four of these by a sum over factorials,
where
and
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 propertiesSee or