Physics
Clebsch Gordan Coefficients
Clebsch-Gordan coefficients are used in quantum mechanics to calculate the total angular momentum of a composite system. They describe the coupling of two angular momenta to form a third, and are used to determine the probability of a particular outcome in a measurement of the total angular momentum.
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4 Key excerpts on "Clebsch Gordan Coefficients"
eBook - PDF- D. B. Lichtenberg(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
CHAPTER 8 CLEBSCH-GORDAN COEFFICIENTS 8.1 Some Properties of the Coefficients Suppose we have the basis vectors of two unitary irreducible representa-tions of a compact simple Lie group. We denote these vectors by φ^ and ψΐ^, where a and β stand for all numbers necessary to specify the first and second representations, and and v stand for all numbers which differentiate among the different states within these representations. Then the basis tensors of the Kronecker product representation are given by ψ^ψ^. In general, these product basis tensors are not the basis tensors of an irreducible representation. However, the basis tensors of any irreducible representation contained in the product can be written as a linear com-bination of the product tensors. The coefficients in the sum are called Clebsch-Gordan coefficients. The irreducible tensors contained in the direct product may be written with a single index φ^ in which case they can be regarded as vectors. Since we shall have occasion to write them in both ways, we shall usually just call them basis functions. Likewise, the product tensors can be written as a linear combination of the irreducible functions. The coefficients in this sum are also called Clebsch-Gordan coefficients. Let us illustrate with SU(2) as an example. In this case, each of the symbols , β, , and v stands for a single number. Then we can write ψ ( » = Σ(«βμν«βΜΨ;Ψ/ ) (8.1) μν 116 8.1 SOME PROPERTIES OF THE COEFFICIENTS 117 where the coefficients (αβμν aßjm) are the SU(2) Clebsch-Gordan co-efficients, often called Wigner coefficients. It is apparant from this expression that φ^ depends on a and /?, since these indices are not summed over. However, it is customary to suppress these indices on φ^ Likewise, the product basis functions can be written as linear combinations of the basis functions of the irreducible representations : W * = («/y« l « # (8.2) jm The coefficients (<χβ]ιηαβμν) are also Clebsch-Gordan coefficients.
eBook - PDF- G.Ya. Lyubarskii(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
Chapter XI Clebsch-Gordan and Racah Coefficients 54. Evaluation of the Clebsch-frordan Coefficients. In sec. 49 it was shown into what irreducible repre-sentations the product L,XD.p of two irreducible re-presentations of the rotation group R decomposes.Now we shall carry out the actual decomposition and shall study the properties of the Clebsch-Gordan coefficients arising from it. Let e ( -j, * m* j-,) be the canonical basis in the space L,. which transforms according to the representa-tion D.^ and f ( -j 2 $ m * 3?^ tiie canonical basis in the space Lp,which transforms according to the re-presentation £.}p· Let us now consider the product of these representa-tions. The basis in the space L-,*L 2 , in which this represen-tation operates.consists of (25-^+1) (2j 2 +l) vectors e m f m ^ ~Jl ^ m l ^ ^l 5 ^ * m 2* ^2) * As snown in sec · the space L-XLp decomposes into a sum of invariant sub-spaces, in which the irreducible representations of weights j( l^-jgl < i <3 1 +i 2 ) operate. We shall de-note by gJ the canonical bases in these subspaces; the superscript denotes the weight of the irreducible repre-sentation that operates in a given subspace. For a fixed m each of the vectors g ffl ( l^-jgl^ J ^ ^3l+d2) is an eigen-vector A in the space ^XLg cor-responding to the eigen value -im. Hence the vectors gj* are linear combinations of those basic vectors e f which are eigen-vectors of A with the same m l m 2 eigen-value,i.e. of e f^ . - 227 - 228 THE THEORY OF GROUPS We put £ϊ» = Σ UJ2m i m 2 jm)e m J mi . / ^ ] The numbers (j^jpiMiu | jm) are called Clebsch-Gordan coefficients· We shall now obtain an explicit expres-sion for these coefficients and shall find some of their properties. The canonical basis g^ is defined,apart from a unimodular factor which depends upon j. Hence (54,1) is insufficient to define the Clebsch-Gordan coef-ficients uniquely. To eliminate this ambiguity we in-troduce the additional condition (ΛΛΛ* 2 |.Μ)>0.
eBook - PDF- Ajit Kumar(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
Chapter 9 Addition of Angular Momenta 9.1 General Theory and the Clebsch–Gordan Coefficients In many problems of interest it is necessary to add angular momenta. For instance, one is required to add the orbital angular momentum, ˆ ~ L, and the spin angular momentum, ˆ ~ S, while studying spin-orbit coupling in atoms. Then, there are problems related to the studies of multi-electron atoms where one has to add two or more orbital angular momenta. Therefore, it is important to discuss the procedure of addition of angular momenta in quantum mechanics. In view of this, in what follows, we shall discuss the general algebraic method for the addition of any two angular momenta. Note that, in this Chapter, we shall write the eigenfunctions of ˆ L 2 in the bra–ket notation as: |‘, mi. Thus, |‘, mi is an eigenvector (or eigenket) of ˆ L 2 with two quantum numbers ‘ and m. If ˆ ~ L happens to be orbital angular momentum, then ‘ represents the orbital quantum number and m stands for the orbital magnetic quantum number. On the other hand, if ˆ ~ L happens to be the spin angular momentum ( ˆ ~ L = ˆ ~ S), then ‘ is spin quantum number i.e., ‘ = s and m equals the spin magnetic quantum number i.e., m = m s . Let us, without specifying the nature, consider the addition of two angular momenta ˆ ~ L 1 and ˆ ~ L 2 : ˆ ~ J = ˆ ~ L 1 + ˆ ~ L 2 . Individually, ˆ ~ L 1 and ˆ ~ L 2 satisfy the following quantum mechanical commutation relations (see Chapter 6): [ ˆ L 1i , ˆ L 1 j ] = i ¯ h ∑ k ε i jk ˆ L 1k , (9.1.1) [ ˆ L 2i , ˆ L 2 j ] = i ¯ h ∑ k ε i jk ˆ L 2k , (9.1.2) where the indices i, j and k take values from 1 to 3. Note that, it is assumed here that ˆ ~ L 1 and ˆ ~ L 2 either correspond to different degrees of freedom, or correspond to the same degree of freedom but belong to different particles. 298
- James D Louck(Author)
- 2011(Publication Date)
- World Scientific(Publisher)
These positive numbers are called Clebsch-Gordan (CG) numbers. They can be generated recur-sively as discussed in Sect. 2.1.1 of [L]. The orthonormal bases (1.16) and (1.20) of the space H j must be related by a unitary transformation A ( j ) of order N ( j ) = producttext n i =1 (2 j i + 1) with (row; column) indices enumerated by ( m ∈ C ( j ); α ,j,m ∈ R ( j )) (see (1.35) below). Thus, we must have the invertible relations: | ( j α ) jm angbracketright = summationdisplay m ∈ C ( j ) parenleftBig A ( j ) tr parenrightBig α ,j,m ; m | j m angbracketright , each α ,j,m ∈ R ( j ) , (1.24) | j m angbracketright = summationdisplay α ,j,m ∈ R ( j ) parenleftBig A ( j ) † parenrightBig α ,j,m ; m | ( j α ) jm angbracketright , each m ∈ C ( j ) . (1.25) Note. We have reversed the role of row and column indices here from that used in [L] (see pp. 87, 90, 91, 94, 95), so that the notation accords with that used later in Chapter 5 for coupling schemes associated with binary trees, and the general structure set forth in Sect. 5.1. square The transformation to a coupled basis (1.20) as given by (1.24) effects the full reduction of the n − fold Kronecker product D j ( U ) = D j 1 ( U ) ⊗ D j 2 ( U ) ⊗ · · · ⊗ D j n ( U ) ,U ∈ SU (2) , (1.26) of SU (2) unitary irreducible matrix representations. The matrix D j ( U ) , U ∈ SU (2) , is a reducible unitary representation of SU (2) of dimension N ( j ) , and the transformation (1.25) effects the transformation to a direct sum of irreducible unitary representations D j ( U ) (Wigner D − matrices). We next summarize the transformation properties of the coupled and ucoupled bases (1.25) under SU (2) frame rotations.
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