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 ...
, the Clebsch–Gordan (CG) coefficients are numbers that arise in
angular momentum coupling 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 chemistry, ...
. They appear as the expansion coefficients of
total angular momentum eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s 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, particularly of
compact Lie groups, 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, who encountered an equivalent problem in
invariant theory.
From a
vector calculus perspective, the CG coefficients associated with the
SO(3) group 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
eigenfunctions of total angular momentum and projection thereof onto an axis, and the integrals correspond to the
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
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
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathem ...
and the
Condon–Shortley phase convention is adopted.
Review of the angular momentum operators
Angular momentum operators are
self-adjoint operators , , 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,
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. 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,
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 i ...
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 m ...
, 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 deltas.
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 (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; 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 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 with ...
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 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
Wigner's 6-''j'' symbols were introduced by Eugene Paul Wigner in 1940 and published in 1965. They are defined as a sum over products of four Wigner 3-j symbols,
:
\begin
j_1 & j_2 & j_3\\
j_4 & j_5 & j_6
\end
= \sum_ (-1)^
\beg ...
*
9-j symbol
*
Racah W-coefficient 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 atomi ...
*
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 m ...
*
Tensor products of representations
*
Associated Legendre polynomials
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation
\left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0,
or equivalently ...
*
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
*
Total angular momentum quantum number
*
Azimuthal quantum number
*
Table of Clebsch–Gordan coefficients
*
Wigner D-matrix
*
Wigner–Eckart theorem
*
Angular momentum diagrams (quantum mechanics)
*
Clebsch–Gordan coefficient for SU(3)
*
Littlewood–Richardson coefficient
Remarks
Notes
References
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*
Albert Messiah (1966). ''Quantum Mechanics'' (Vols. I & II), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons.
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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