Clebsch-Gordan coefficients: Difference between revisions
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==Clebsch-Gordan coefficients== | ==Clebsch-Gordan coefficients== | ||
Although '''Clebsch-Gordan coefficients''' can be defined for arbitrary groups, we restrict attention in this article to the groups associated with space and spin angular momentum, namely the groups SO(3) and SU(2). In that case CG coefficients can be defined as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. | Although '''Clebsch-Gordan coefficients''' can be defined for arbitrary groups, we restrict our attention in this article to the groups associated with space and spin angular momentum, namely the groups SO(3) and SU(2). In that case CG coefficients can be defined as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. | ||
Below, this definition is made precise by defining angular momentum | Below, this definition is made precise by defining angular momentum | ||
operators, angular momentum eigenstates, and tensor products of these states. | operators, angular momentum eigenstates, and tensor products of these states. |
Revision as of 06:25, 20 August 2007
In quantum mechanics, the Clebsch-Gordan coefficients (CG coefficients) are sets of numbers that arise in angular momentum coupling.
In mathematics, the CG coefficients appear in group representation theory, particularly of compact Lie groups. They arise in the explicit direct sum decomposition of the outer product of two irreducible representations (irreps) of a group G. In general the outer product representation (rep)—which is carried by a tensor product space—is reducible under G. Decomposition of the outer product rep into irreps of G requires a basis transformation of the tensor product space. The CG coefficients are the elements of the matrix of this basis transformation. In physics it is common to consider only orthonormal bases of the vector spaces involved, and then CG coefficients constitute a unitary matrix.
The name derives from the German mathematicians Alfred Clebsch (1833-1872) and Paul Gordan (1837-1912), who encountered an equivalent problem in invariant theory.
The formulas below use Dirac's bra-ket notation, i.e., the quantity stands for a positive definite inner product between the elements ψ and φ of the same complex inner product space. We follow the physical convention , where is the complex conjugate of the complex number c.
Clebsch-Gordan coefficients
Although Clebsch-Gordan coefficients can be defined for arbitrary groups, we restrict our attention in this article to the groups associated with space and spin angular momentum, namely the groups SO(3) and SU(2). In that case CG coefficients can be defined as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. Below, this definition is made precise by defining angular momentum operators, angular momentum eigenstates, and tensor products of these states.
From the formal definition recursion relations for the Clebsch-Gordan coefficients can be found. In order to settle the numerical values for the coefficients, a phase convention must be adopted. Below the Condon and Shortley phase convention is chosen.
Angular momentum operators
Angular momentum operators are Hermitian operators , and that satisfy the commutation relations
where is the Levi-Civita symbol. Together the three components define a vector operator . The square of the length of is defined as
We also define raising and lowering operators
Angular momentum states
It can be shown from the above definitions that commutes with and
When two Hermitian operators commute a common set of eigenfunctions exists. Conventionally and are chosen. From the commutation relations the possible eigenvalues can be found. The result is
The raising and lowering operators change the value of
with
A (complex) phase factor could be included in the definition of The choice made here is in agreement with the Condon and Shortley phase conventions. The angular momentum states must be orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and they are assumed to be normalized
Tensor product space
Let be the dimensional vector space spanned by the states
and the dimensional vector space spanned by
The tensor product of these spaces, , has a dimensional uncoupled basis
Angular momentum operators acting on can be defined by
and
Total angular momentum operators are defined by
The total angular momentum operators satisfy the required commutation relations
and hence total angular momentum eigenstates exist
It can be derived that must satisfy the triangular condition
The total number of total angular momentum eigenstates is equal to the dimension of
The total angular momentum states form an orthonormal basis of
Formal definition of Clebsch-Gordan coefficients
The total angular momentum states can be expanded in the uncoupled basis
The expansion coefficients are called Clebsch-Gordan coefficients.
Applying the operator
to both sides of the defining equation shows that the Clebsch-Gordan coefficients can only be nonzero when
Recursion relations
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
Combining these results gives recursion relations for the Clebsch-Gordan coefficients
Taking the upper sign with gives
In the Condon and Shortley phase convention the coefficient is taken real and positive. With the last equation all other Clebsch-Gordan coefficients can be found. The normalization is fixed by the requirement that the sum of the squares, which corresponds to 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 and Shortley phase convention).
Explicit expression
For an explicit expression of the Clebsch-Gordan coefficients and tables with numerical values see table of Clebsch-Gordan coefficients.
Orthogonality relations
These are most clearly written down by introducing the alternative notation
The first orthogonality relation is
and the second
Special cases
For the Clebsch-Gordan coefficients are given by
For and we have
Symmetry properties
Relation to 3-jm symbols
Clebsch-Gordan coefficients are related to 3-jm symbols which have more convenient symmetry relations.
See also
- 3-jm symbol
- Racah W-coefficient
- 6-j symbol
- 9-j symbol
- Spherical harmonics
- Associated Legendre polynomials
- Angular momentum
- Angular momentum coupling
- Total electronic angular momentum quantum number
- Azimuthal quantum number
- Table of Clebsch-Gordan coefficients
External links
References
- Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 0-691-07912-9.
- Condon, Edward U.; Shortley, G. H. (1970). “Chapter 3”, The Theory of Atomic Spectra. Cambridge: Cambridge University Press. ISBN 0-521-09209-4.
- Messiah, Albert (1981). Quantum Mechanics (Volume II), 12th edition. New York: North Holland Publishing. ISBN 0-7204-0045-7.
- Brink, D. M.; Satchler, G. R. (1993). “Chapter 2”, Angular Momentum, 3rd edition. Oxford: Clarendon Press. ISBN 0-19-851759-9.
- Zare, Richard N. (1988). “Chapter 2”, Angular Momentum. New York: John Wiley & Sons. ISBN 0-471-85892-7.
- Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading, Massachusetts: Addison-Wesley. ISBN 0201135078.