In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the Clebsch–Gordan (CG) coefficients are numbers that arise in
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 ...
in
quantum mechanics
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 chemistr ...
. They appear as the expansion coefficients of
total angular momentum
In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).
If s is the particle's sp ...
eigenstates in an uncoupled
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
basis. In more mathematical terms, the CG coefficients are used in
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, particularly of
compact Lie group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
s, to perform the explicit
direct sum decomposition of the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of two
irreducible representations (i.e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly). The name derives from the German mathematicians
Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...
and
Paul Gordan
__NOTOC__
Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German mathematician, a student of Carl Jacobi at the University of Königsberg before obtaining his PhD at the University of Breslau (1862),. and a professor ...
, who encountered an equivalent problem in
invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
.
From a
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
perspective, the CG coefficients associated with the
SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
can be defined simply in terms of integrals of products of
spherical harmonic
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form ...
s and their complex conjugates. The addition of spins in quantum-mechanical terms can be read directly from this approach as spherical harmonics are
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of total angular momentum and projection thereof onto an axis, and the integrals correspond to the
Hilbert space inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. There also exist complicated explicit formulas for their direct calculation.
The formulas below use
Dirac's bra–ket notation and the
Condon–Shortley phase convention is adopted.
Review of the angular momentum operators
Angular momentum operators are
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
s , , and that satisfy the
commutation relation
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
s
where is the
Levi-Civita symbol. Together the three operators define a ''vector operator'', a rank one Cartesian
tensor operator
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the ...
,
It is also known as a
spherical vector, since it is also a spherical tensor operator. It is only for rank one that spherical tensor operators coincide with the Cartesian tensor operators.
By developing this concept further, one can define another operator as the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of with itself:
This is an example of a
Casimir operator
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
. It is diagonal and its eigenvalue characterizes the particular
irreducible representation of the angular momentum algebra
. This is physically interpreted as the square of the total angular momentum of the states on which the representation acts.
One can also define ''raising'' () and ''lowering'' () operators, the so-called
ladder operators
In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
,
Spherical basis for angular momentum eigenstates
It can be shown from the above definitions that commutes with , , and :
When two
Hermitian operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to it ...
s commute, a common set of eigenstates exists. Conventionally, and are chosen. From the commutation relations, the possible eigenvalues can be found. These eigenstates are denoted where is the ''angular momentum quantum number'' and is the ''angular momentum projection'' onto the z-axis.
They comprise the
spherical basis
In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular mo ...
, are complete, and satisfy the following eigenvalue equations,
The raising and lowering operators can be used to alter the value of ,
where the ladder coefficient is given by:
In principle, one may also introduce a (possibly complex) phase factor in the definition of
. The choice made in this article is in agreement with the
Condon–Shortley phase convention. The angular momentum states are orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and are assumed to be normalized,
Here the italicized and denote integer or half-integer
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 ...
quantum numbers of a particle or of a system. On the other hand, the roman , , , , , and denote operators. The
symbols are
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
s.
Tensor product space
We now consider systems with two physically different angular momenta and . Examples include the spin and the orbital angular momentum of a single electron, or the spins of two electrons, or the orbital angular momenta of two electrons. Mathematically, this means that the angular momentum operators act on a space
of dimension
and also on a space
of dimension
. We are then going to define a family of "total angular momentum" operators acting on the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
space
, which has dimension
. The action of the total angular momentum operator on this space constitutes a representation of the su(2) Lie algebra, but a reducible one. The reduction of this reducible representation into irreducible pieces is the goal of Clebsch–Gordan theory.
Let be the -dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
spanned by the states
and the -dimensional vector space spanned by the states
The tensor product of these spaces, , has a -dimensional ''uncoupled'' basis
Angular momentum operators are defined to act on states in in the following manner:
and
where denotes the identity operator.
The total
[The word "total" is often overloaded to mean several different things. In this article, "total angular momentum" refers to a generic sum of two angular momentum operators and . It is not to be confused with the other common use of the term "total angular momentum" that refers specifically to the sum of orbital angular momentum and spin.] angular momentum operators are defined by the
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
(or
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
) of the two representations acting on ,
The total angular momentum operators can be shown to ''satisfy the very same commutation relations'',
where . Indeed, the preceding construction is the standard method for constructing an action of a Lie algebra on a tensor product representation.
Hence, a set of ''coupled'' eigenstates exist for the total angular momentum operator as well,
for . Note that it is common to omit the part.
The total angular momentum quantum number must satisfy the triangular condition that
such that the three nonnegative integer or half-integer values could correspond to the three sides of a triangle.
The total number of total angular momentum eigenstates is necessarily equal to the dimension of :
As this computation suggests, the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension
, where
ranges from
to
in increments of 1. As an example, consider the tensor product of the three-dimensional representation corresponding to
with the two-dimensional representation with
. The possible values of
are then
and
. Thus, the six-dimensional tensor product representation decomposes as the direct sum of a two-dimensional representation and a four-dimensional representation.
The goal is now to describe the preceding decomposition explicitly, that is, to explicitly describe basis elements in the tensor product space for each of the component representations that arise.
The total angular momentum states form an orthonormal basis of :
These rules may be iterated to, e.g., combine doublets (=1/2) to obtain the Clebsch-Gordan decomposition series, (
Catalan's triangle),
where
is the integer
floor function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
; and the number preceding the boldface irreducible representation dimensionality () label indicates multiplicity of that representation in the representation reduction. For instance, from this formula, addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s,
.
Formal definition of Clebsch–Gordan coefficients
The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis
The expansion coefficients
are the ''Clebsch–Gordan coefficients''. Note that some authors write them in a different order such as . Another common notation is
.
Applying the operators
to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when
Recursion relations
The recursion relations were discovered by physicist
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 ...
from the Hebrew University of Jerusalem in 1941.
Applying the total angular momentum raising and lowering operators
to the left hand side of the defining equation gives
Applying the same operators to the right hand side gives
where was defined in . Combining these results gives recursion relations for the Clebsch–Gordan coefficients:
Taking the upper sign with the condition that gives initial recursion relation:
In the Condon–Shortley phase convention, one adds the constraint that
:
(and is therefore also real).
The Clebsch–Gordan coefficients can then be found from these recursion relations. The normalization is fixed by the requirement that the sum of the squares, which equivalent to the requirement that the norm of the state must be one.
The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with . Repeated use of that equation gives all coefficients.
This procedure to find the Clebsch–Gordan coefficients shows that they are all real in the Condon–Shortley phase convention.
Explicit expression
Orthogonality relations
These are most clearly written down by introducing the alternative notation
The first orthogonality relation is
(derived from the fact that
) and the second one is
Special cases
For the Clebsch–Gordan coefficients are given by
For and we have
For and we have
For we have
For , we have
For we have
Symmetry properties
A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to
Wigner 3-j symbols using . The symmetry properties of Wigner 3-j symbols are much simpler.
Rules for phase factors
Care is needed when simplifying phase factors: a quantum number may be a half-integer rather than an integer, therefore is not necessarily for a given quantum number unless it can be proven to be an integer. Instead, it is replaced by the following weaker rule:
for any angular-momentum-like quantum number .
Nonetheless, a combination of and is always an integer, so the stronger rule applies for these combinations:
This identity also holds if the sign of either or or both is reversed.
It is useful to observe that any phase factor for a given pair can be reduced to the canonical form:
where and (other conventions are possible too). Converting phase factors into this form makes it easy to tell whether two phase factors are equivalent. (Note that this form is only ''locally'' canonical: it fails to take into account the rules that govern combinations of pairs such as the one described in the next paragraph.)
An additional rule holds for combinations of , , and that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol:
This identity also holds if the sign of any is reversed, or if any of them are substituted with an instead.
Relation to Wigner 3-j symbols
Clebsch–Gordan coefficients are related to
Wigner 3-j symbols which have more convenient symmetry relations.
The factor is due to the Condon–Shortley constraint that , while is due to the time-reversed nature of .
Relation to Candy matrices
Relation to spherical harmonics
In the case where integers are involved, the coefficients can be related to
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
s of
spherical harmonic
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form ...
s:
It follows from this and orthonormality of the spherical harmonics that CG coefficients are in fact the expansion coefficients of a product of two spherical harmonics in terms of a single spherical harmonic:
Other Properties
Clebsch–Gordan coefficients for specific groups
For arbitrary groups and their representations, Clebsch–Gordan coefficients are not known in general. However, algorithms to produce Clebsch–Gordan coefficients for the
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
SU(''n'') are known. In particular,
SU(3) Clebsch-Gordan coefficients have been computed and tabulated because of their utility in characterizing hadronic decays, where a
flavor-SU(3) symmetry exists that relates the
up,
down, and
strange quarks.
web interface for tabulating SU(N) Clebsch–Gordan coefficientsis readily available.
Clebsch–Gordan coefficients for symmetric group are also known as
Kronecker coefficients.
See also
*
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
*
9-j symbol
In physics, Wigner's 9-''j'' symbols were introduced by Eugene Paul Wigner in 1937. They are related to recoupling coefficients in quantum mechanics involving four angular momenta
\sqrt
\begin
j_1 & j_2 & j_3\\
j_4 & j_5 & j_6\\
...
*
Racah W-coefficient
*
Spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form ...
*
Spherical basis
In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular mo ...
*
Tensor products of representations
*
Associated Legendre polynomials
*
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 ...
*
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 ...
*
Total angular momentum quantum number
In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).
If s is the particle's sp ...
*
Azimuthal quantum number
The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe ...
*
Table of Clebsch–Gordan coefficients
*
Wigner D-matrix The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex con ...
*
Wigner–Eckart theorem
The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators in the basis of angular momentum eigenstates can be expressed as the product of two factors, on ...
*
Angular momentum diagrams (quantum mechanics)
In quantum mechanics and its applications to quantum many-particle systems, notably quantum chemistry, angular momentum diagrams, or more accurately from a mathematical viewpoint angular momentum graphs, are a diagrammatic method for representing ...
*
Clebsch–Gordan coefficient for SU(3)
*
Littlewood–Richardson coefficient
Remarks
Notes
References
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Albert Messiah
Albert Messiah (23 September 1921, Nice – 17 April 2013, Paris) was a French physicist.
He studied at the Ecole Polytechnique.
He spent the Second World War in the Free France forces: he embarked on 22 June 1940 at Saint-Jean-de-Luz for Engla ...
(1966). ''Quantum Mechanics'' (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
*
External links
*
Clebsch–Gordan, 3-j and 6-j Coefficient Web CalculatorDownloadable Clebsch–Gordan Coefficient Calculator for Mac and WindowsWeb interface for tabulating SU(N) Clebsch–Gordan coefficients
Further reading
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{{DEFAULTSORT:Clebsch-Gordan coefficients
Rotation in three dimensions
Rotational symmetry
Representation theory of Lie groups
Quantum mechanics
Mathematical physics