Racah's W-coefficients were introduced by
Giulio Racah
Giulio (Yoel) Racah ( he, ג'וליו (יואל) רקח; February 9, 1909 – August 28, 1965) was an Italian–Israeli physicist and mathematician. He was Acting President of the Hebrew University of Jerusalem from 1961 to 1962.
The crater ...
in 1942. These coefficients have a purely mathematical definition. In physics they are used in calculations involving the
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
description of
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, for example in
atomic theory
Atomic theory is the scientific theory that matter is composed of particles called atoms. Atomic theory traces its origins to an ancient philosophical tradition known as atomism. According to this idea, if one were to take a lump of matter a ...
.
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
In atomic theory and quantum mechanics, an atomic orbital is a Function (mathematics), function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electr ...
. Each electron has
electron spin
In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin (physics), spin and electric charge. The value of the ...
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
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 t ...
. 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 w coefficient](https://upload.wikimedia.org/wikipedia/commons/2/21/Racah_w_coefficient.svg)
Racah coefficients are related to recoupling coefficients by
:
Recoupling coefficients are elements of a
unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
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 coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In ...
. 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
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
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
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an Operator (mathematics), operator that increases or decreases the eigenvalue of another operator. In quantu ...
to both sides of the equation.
Algebra
Let
:
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
:
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
:
Cf.
[Brink, D M & Satchler, G R (1968). ''Angular Momentum'' (Oxford University Press) 3 ed., p. 142]
online
/ref> or
:
See also
* Clebsch–Gordan coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In ...
* 3-j symbol In quantum mechanics, the Wigner 3-j symbols, also called 3''-jm'' symbols, are an alternative to Clebsch–Gordan coefficients for the purpose of adding angular momenta. While the two approaches address exactly the same physical problem, the 3-''j' ...
* 6-j symbol
* Pandya theorem
Notes
Further reading
*
*
*
*
*
*
*
External links
*
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Rotational symmetry
Representation theory of Lie groups