Clebsch–Gordan coefficients
In physics, the Clebsch–Gordan 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 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 of two irreducible representations. The name derives from the German mathematicians Alfred Clebsch and Paul Gordan, who encountered an equivalent problem in invariant theory.
From a vector calculus perspective, the CG coefficients associated with the SO group can be defined simply in terms of integrals of products of spherical harmonics 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 inner product. 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.
Angular momentum operators
Angular momentum operators are self-adjoint operators,, and that satisfy the commutation relationswhere is the Levi-Civita symbol. Together the three operators define a vector operator, a rank one Cartesian tensor operator,
It 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 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 operators 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, 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 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 and are assumed to be normalized,
Here the italicized denote integer or half-integer angular momentum 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 space, which has dimension. The action of the total angular momentum operator on this space constitutes a representation of the su 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 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 angular momentum operators are defined by the coproduct 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 to obtain the Clebsch-Gordan decomposition series,,
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 in the uncoupled basisThe 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 operator
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
.
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 notationThe first orthogonality relation is
and the second one is
Special cases
For the Clebsch–Gordan coefficients are given byFor 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 . Converting phase factors into this form makes it easy to tell whether two phase factors are equivalent.
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 Wigner D-matrices
Relation to spherical harmonics
In the case where integers are involved, the coefficients can be related to integrals of spherical harmonics: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: