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 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 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 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 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.[1] 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.[2]
The formulas below use Dirac's bra–ket notation and the Condon–Shortley phase convention[3] is adopted.
Angular momentum operators
Angular momentum operators are self-adjoint operators jx, jy, and jz that satisfy the commutation relations
where εklm 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 j2 as the inner product of j 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 so(3) ≅ su(2). 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 (j+) and lowering (j−) operators, the so-called ladder operators,
Angular momentum states
It can be shown from the above definitions that j2 commutes with jx, jy, and jz:
When two Hermitian operators commute, a common set of eigenfunctions exists. Conventionally j2 and jz are chosen. From the commutation relations the possible eigenvalues can be found. These states are denoted |j m⟩ where j is the angular momentum quantum number and m is the angular momentum projection onto the z-axis. They satisfy the following eigenvalue equations:
The raising and lowering operators can be used to alter the value of m:
where the ladder coefficient is given by:
-
(1)
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 j and m denote integer or half-integer angular momentum quantum numbers of a particle or of a system. On the other hand, the roman jx, jy, jz, j+, j−, and j2 denote operators. The symbols are Kronecker deltas.
Tensor product space
We now consider systems with two physically different angular momenta j1 and j2. 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.
Let V1 be the (2 j1 + 1)-dimensional vector space spanned by the states
- ,
and V2 the (2 j2 + 1)-dimensional vector space spanned by the states
- .
The tensor product of these spaces, V3 ≡ V1 ⊗ V2, has a (2 j1 + 1) (2 j2 + 1)-dimensional uncoupled basis
- .
Angular momentum operators are defined to act on states in V3 in the following manner:
and
where 1 denotes the identity operator.
The total[nb 1] angular momentum operators are defined by
The total angular momentum operators can be shown to satisfy the same kind of commutation relations:
- ,
where k, l, m ∈ {x, y, z}. Hence, a set of coupled eigenstates exist for the total angular momentum operator as well:
for M ∈ {−J, −J + 1, …, J}. Note that it is common to omit the [j1 j2] part.
The total angular momentum quantum number J 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.[4]
The total number of total angular momentum eigenstates is necessarily equal to the dimension of V3:
- .
The total angular momentum states form an orthonormal basis of V3:
- .
Formal definition of Clebsch–Gordan coefficients
The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis
-
(2)
The expansion coefficients
are the Clebsch–Gordan coefficients. Note that some authors write them in a different order such as ⟨j1 j2; m1 m2|J M⟩.
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 C± was defined in 1. Combining these results gives recursion relations for the Clebsch–Gordan coefficients:
- .
Taking the upper sign with the condition that M = J gives initial recursion relation:
- .
In the Condon–Shortley phase convention, one adds the constraint that
(and is therefore also real).
The Clebsch–Gordan coefficients ⟨j1 m1 j2 m2 | J J⟩ 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 |[j1 j2] J J⟩ must be one.
The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with M = J − 1. 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 1 ≡ ∑x |x⟩ ⟨x|) and the second one is
- .
Special cases
For J = 0 the Clebsch–Gordan coefficients are given by
- .
For J = j1 + j2 and M = J we have
- .
For j1 = j2 = J / 2 and m1 = −m2 we have
- .
For j1 = j2 = m1 = −m2 we have
For j2 = 1, m2 = 0 we have
Symmetry properties
A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3-j symbols using 3. 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 (−1)2k is not necessarily 1 for a given quantum number k 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 k.
Nonetheless, a combination of ji and mi is always an integer, so the stronger rule applies for these combinations:
This identity also holds if the sign of either ji or mi or both is reversed.
It is useful to observe that any phase factor for a given (ji, mi) pair can be reduced to the canonical form:
where a ∈ {0, 1, 2, 3} and b ∈ {0, 1} (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 (ji, mi) pairs such as the one described in the next paragraph.)
An additional rule holds for combinations of j1, j2, and j3 that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol:
This identity also holds if the sign of any ji is reversed, or if any of them are substituted with an mi instead.
Relation to Wigner 3-j symbols
Clebsch–Gordan coefficients are related to Wigner 3-j symbols which have more convenient symmetry relations.
-
(3)
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 a single spherical harmonic:
Other Properties
SU(N) Clebsch–Gordan coefficients
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 are known.[5][6] 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.[7][8] A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available.
See also
- 3-jm symbol
- Racah W-coefficient
- 6-j symbol
- 9-j symbol
- Spherical harmonics
- Spherical basis
- Associated Legendre polynomials
- Angular momentum
- Angular momentum coupling
- Total angular momentum quantum number
- Azimuthal quantum number
- Table of Clebsch–Gordan coefficients
- Wigner D-matrix
- Angular momentum diagrams (quantum mechanics)
- Clebsch–Gordan coefficient for SU(3)
Remarks
- ↑ 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 j1 and j2. 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.
Notes
References
- Alex, A.; Kalus, M.; Huckleberry, A.; von Delft, J. (2011). "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients". J. Math. Phys. 82: 023507. arXiv:1009.0437. Bibcode:2011JMP....52b3507A. doi:10.1063/1.3521562.
- Condon, Edward U.; Shortley, G. H. (1970). "Ch. 3". The Theory of Atomic Spectra. Cambridge: Cambridge University Press. ISBN 0-521-09209-4.
- Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 0-691-07912-9.
- Greiner, Walter; Müller, Berndt (1994). Quantum Mechanics: Symmetries (2nd ed.). Springer Verlag. ISBN 978-3540580805.
- Kaplan, L. M.; Resnikoff, M. (1967). "Matrix products and explicit 3, 6, 9, and 12j coefficients of the regular representation of SU(n)". J. Math. Phys. 8: 2194. Bibcode:1967JMP.....8.2194K. doi:10.1063/1.1705141.
- Kaeding, Thomas (1995). "Tables of SU(3) isoscalar factors". arXiv:nucl-th/9502037 [nucl-th].
- Merzbacher, Eugen (1998). Quantum Mechanics (3rd ed.). John Wiley. pp. 428–9. ISBN 0-471-88702-1.
- de Swart, J. J. (1963). "The Octet model and its Clebsch-Gordan coefficients". Rev. Mod. Phys. 35: 916. Bibcode:1963RvMP...35..916D. doi:10.1103/RevModPhys.35.916.
External links
- PDF Table of Clebsch–Gordan Coefficients, Spherical Harmonics, and d-Functions
- Clebsch–Gordan, 3-j and 6-j Coefficient Web Calculator
- Downloadable Clebsch–Gordan Coefficient Calculator for Mac and Windows
- Web interface for tabulating SU(N) Clebsch–Gordan coefficients
Further reading
- Quantum mechanics, E. Zaarur, Y. Peleg, R. Pnini, Schaum's Easy Oulines Crash Course, McGraw Hill (USA), 2006, ISBN (10-)007-145533-7 ISBN (13-)978-007-145533-6
- Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition), R. Eisberg, R. Resnick, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
- Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
- Physics of Atoms and Molecules, B. H. Bransden, C. J. Joachain, Longman, 1983, ISBN 0-582-44401-2
- The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
- Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
- McGraw Hill Encyclopaedia of Physics (2nd Edition), C. B. Parker, 1994, ISBN 0-07-051400-3
- Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading, Massachusetts: Addison-Wesley. ISBN 0-201-13507-8.
- Brink, D. M.; Satchler, G. R. (1993). "Ch. 2". Angular Momentum (3rd ed.). Oxford: Clarendon Press. ISBN 0-19-851759-9.
- Messiah, Albert (1981). "Ch. XIII". Quantum Mechanics (Volume II). New York: North Holland Publishing. ISBN 0-7204-0045-7.
- Zare, Richard N. (1988). "Ch. 2". Angular Momentum. New York: John Wiley & Sons. ISBN 0-471-85892-7.
|