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.^[1] While the two approaches address exactly the same physical problem, the 3-j symbols do so more symmetrically.

Mathematical relation to Clebsch–Gordan coefficients

The 3-j symbols are given in terms of the Clebsch–Gordan coefficients by

(\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) \equiv \frac{(- 1)^{j_{1} - j_{2} - m_{3}}}{\sqrt{2 j_{3} + 1}} ⟨ j_{1} m_{1} j_{2} m_{2} | j_{3} (- m_{3}) ⟩ .

The j and m components are angular-momentum quantum numbers, i.e., every $j$ (and every corresponding $m$ ) is either a nonnegative integer or half-odd-integer. The exponent of the sign factor is always an integer, so it remains the same when transposed to the left, and the inverse relation follows upon making the substitution $m 3 \to - m 3$ :

⟨ j_{1} m_{1} j_{2} m_{2} | j_{3} m_{3} ⟩ = (- 1)^{- j_{1} + j_{2} - m_{3}} \sqrt{2 j_{3} + 1} (\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & - m_{3} \end{matrix}) .

Definitional relation to Clebsch–Gordan coefficients

The CG coefficients are defined so as to express the addition of two angular momenta in terms of a third:

| j_{3} m_{3} ⟩ = \sum_{m_{1} = - j_{1}}^{j_{1}} \sum_{m_{2} = - j_{2}}^{j_{2}} ⟨ j_{1} m_{1} j_{2} m_{2} | j_{3} m_{3} ⟩ | j_{1} m_{1} j_{2} m_{2} ⟩ .

The 3-j symbols, on the other hand, are the coefficients with which three angular momenta must be added so that the resultant is zero:

\sum_{m_{1} = - j_{1}}^{j_{1}} \sum_{m_{2} = - j_{2}}^{j_{2}} \sum_{m_{3} = - j_{3}}^{j_{3}} | j_{1} m_{1} ⟩ | j_{2} m_{2} ⟩ | j_{3} m_{3} ⟩ (\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) = | 0 0 ⟩ .

Here $| 0 0 ⟩$ is the zero-angular-momentum state ( $j = m = 0$ ). It is apparent that the 3-j symbol treats all three angular momenta involved in the addition problem on an equal footing and is therefore more symmetrical than the CG coefficient.

Since the state $| 0 0 ⟩$ is unchanged by rotation, one also says that the contraction of the product of three rotational states with a 3-j symbol is invariant under rotations.

Selection rules

The Wigner 3-j symbol is zero unless all these conditions are satisfied:

\begin{aligned} m_{i} \in {- j_{i}, - j_{i} + 1, - j_{i} + 2, \dots, j_{i}} (i = 1, 2, 3), \\ m_{1} + m_{2} + m_{3} = 0, \\ | j_{1} - j_{2} | \leq j_{3} \leq j_{1} + j_{2}, \\ (j_{1} + j_{2} + j_{3}) is an integer (and, moreover, an even integer if m_{1} = m_{2} = m_{3} = 0) . \end{aligned}

Symmetry properties

A 3-j symbol is invariant under an even permutation of its columns:

(\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) = (\begin{matrix} j_{2} & j_{3} & j_{1} \\ m_{2} & m_{3} & m_{1} \end{matrix}) = (\begin{matrix} j_{3} & j_{1} & j_{2} \\ m_{3} & m_{1} & m_{2} \end{matrix}) .

An odd permutation of the columns gives a phase factor:

(\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) = (- 1)^{j_{1} + j_{2} + j_{3}} (\begin{matrix} j_{2} & j_{1} & j_{3} \\ m_{2} & m_{1} & m_{3} \end{matrix}) = (- 1)^{j_{1} + j_{2} + j_{3}} (\begin{matrix} j_{1} & j_{3} & j_{2} \\ m_{1} & m_{3} & m_{2} \end{matrix}) = (- 1)^{j_{1} + j_{2} + j_{3}} (\begin{matrix} j_{3} & j_{2} & j_{1} \\ m_{3} & m_{2} & m_{1} \end{matrix}) .

Changing the sign of the $m$ quantum numbers (time reversal) also gives a phase:

(\begin{matrix} j_{1} & j_{2} & j_{3} \\ - m_{1} & - m_{2} & - m_{3} \end{matrix}) = (- 1)^{j_{1} + j_{2} + j_{3}} (\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) .

The 3-j symbols also have so-called Regge symmetries, which are not due to permutations or time reversal.^[2] These symmetries are:

(\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) = (\begin{matrix} j_{1} & \frac{j_{2} + j_{3} - m_{1}}{2} & \frac{j_{2} + j_{3} + m_{1}}{2} \\ j_{3} - j_{2} & \frac{j_{2} - j_{3} - m_{1}}{2} - m_{3} & \frac{j_{2} - j_{3} + m_{1}}{2} + m_{3} \end{matrix}),

(\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) = (- 1)^{j_{1} + j_{2} + j_{3}} (\begin{matrix} \frac{j_{2} + j_{3} + m_{1}}{2} & \frac{j_{1} + j_{3} + m_{2}}{2} & \frac{j_{1} + j_{2} + m_{3}}{2} \\ j_{1} - \frac{j_{2} + j_{3} - m_{1}}{2} & j_{2} - \frac{j_{1} + j_{3} - m_{2}}{2} & j_{3} - \frac{j_{1} + j_{2} - m_{3}}{2} \end{matrix}) .

With the Regge symmetries, the 3-j symbol has a total of 72 symmetries. These are best displayed by the definition of a Regge symbol, which is a one-to-one correspondence between it and a 3-j symbol and assumes the properties of a semi-magic square:^[3]

R = \begin{array}{ccc} - j_{1} + j_{2} + j_{3} & j_{1} - j_{2} + j_{3} & j_{1} + j_{2} - j_{3} \\ j_{1} - m_{1} & j_{2} - m_{2} & j_{3} - m_{3} \\ j_{1} + m_{1} & j_{2} + m_{2} & j_{3} + m_{3} \end{array},

whereby the 72 symmetries now correspond to 3! row and 3! column interchanges plus a transposition of the matrix. These facts can be used to devise an effective storage scheme.^[4]

Orthogonality relations

A system of two angular momenta with magnitudes $j 1$ and $j 2$ can be described either in terms of the uncoupled basis states (labeled by the quantum numbers $m 1$ and $m 2$ ), or the coupled basis states (labeled by $j 3$ and $m 3$ ). The 3-j symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality relations

(2 j_{3} + 1) \sum_{m_{1} m_{2}} (\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) (\begin{matrix} j_{1} & j_{2} & j'_{3} \\ m_{1} & m_{2} & m'_{3} \end{matrix}) = δ_{j_{3}, j'_{3}} δ_{m_{3}, m'_{3}} {\begin{matrix} j_{1} & j_{2} & j_{3} \end{matrix}},

\sum_{j_{3} m_{3}} (2 j_{3} + 1) (\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) (\begin{matrix} j_{1} & j_{2} & j_{3} \\ {m_{1}}^{'} & {m_{2}}^{'} & m_{3} \end{matrix}) = δ_{m_{1}, {m_{1}}^{'}} δ_{m_{2}, {m_{2}}^{'}} .

The triangular delta ${j 1 j 2 j 3}$ is equal to 1 when the triad (j₁, j₂, j₃) satisfies the triangle conditions, and is zero otherwise. The triangular delta itself is sometimes confusingly called^[5] a "3-j symbol" (without the m) in analogy to 6-j and 9-j symbols, all of which are irreducible summations of 3-jm symbols where no $m$ variables remain.

Relation to spherical harmonics

The 3-jm symbols give the integral of the products of three spherical harmonics^{[citation needed]}

\begin{aligned} \int Y_{l_{1} m_{1}} (θ, φ) Y_{l_{2} m_{2}} (θ, φ) Y_{l_{3} m_{3}} (θ, φ) \sin θ d θ d φ \\ = \sqrt{\frac{(2 l_{1} + 1) (2 l_{2} + 1) (2 l_{3} + 1)}{4 π}} (\begin{matrix} l_{1} & l_{2} & l_{3} \\ 0 & 0 & 0 \end{matrix}) (\begin{matrix} l_{1} & l_{2} & l_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) \end{aligned}

with $l_{1}$ , $l_{2}$ and $l_{3}$ integers.

Relation to integrals of spin-weighted spherical harmonics

Similar relations exist for the spin-weighted spherical harmonics if $s_{1} + s_{2} + s_{3} = 0$ :

\begin{aligned} \int d \hat{n}_{s_{1}} Y_{j_{1} m_{1}} (\hat{n})_{s_{2}} Y_{j_{2} m_{2}} (\hat{n})_{s_{3}} Y_{j_{3} m_{3}} (\hat{n}) \\ = \sqrt{\frac{(2 j_{1} + 1) (2 j_{2} + 1) (2 j_{3} + 1)}{4 π}} (\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) (\begin{matrix} j_{1} & j_{2} & j_{3} \\ - s_{1} & - s_{2} & - s_{3} \end{matrix}) . \end{aligned}

Recursion relations

\begin{aligned} - \sqrt{(l_{3} \mp s_{3}) (l_{3} \pm s_{3} + 1)} (\begin{matrix} l_{1} & l_{2} & l_{3} \\ s_{1} & s_{2} & s_{3} \pm 1 \end{matrix}) = \\ = \sqrt{(l_{1} \mp s_{1}) (l_{1} \pm s_{1} + 1)} (\begin{matrix} l_{1} & l_{2} & l_{3} \\ s_{1} \pm 1 & s_{2} & s_{3} \end{matrix}) + \sqrt{(l_{2} \mp s_{2}) (l_{2} \pm s_{2} + 1)} (\begin{matrix} l_{1} & l_{2} & l_{3} \\ s_{1} & s_{2} \pm 1 & s_{3} \end{matrix}) . \end{aligned}

Asymptotic expressions

For $l_{1} ≪ l_{2}, l_{3}$ a non-zero 3-j symbol is

(\begin{matrix} l_{1} & l_{2} & l_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) \approx (- 1)^{l_{3} + m_{3}} \frac{d_{m_{1}, l_{3} - l_{2}}^{l_{1}} (θ)}{\sqrt{2 l_{3} + 1}},

where $\cos (θ) = - 2 m_{3} / (2 l_{3} + 1)$ , and $d_{m n}^{l}$ is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by

(\begin{matrix} l_{1} & l_{2} & l_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) \approx (- 1)^{l_{3} + m_{3}} \frac{d_{m_{1}, l_{3} - l_{2}}^{l_{1}} (θ)}{\sqrt{l_{2} + l_{3} + 1}},

where $\cos (θ) = (m_{2} - m_{3}) / (l_{2} + l_{3} + 1)$ .

Metric tensor

The following quantity acts as a metric tensor in angular-momentum theory and is also known as a Wigner 1-jm symbol:^[1]

(\begin{matrix} j \\ m m^{'} \end{matrix}) : = \sqrt{2 j + 1} (\begin{matrix} j & 0 & j \\ m & 0 & m^{'} \end{matrix}) = (- 1)^{j - m^{'}} δ_{m, - m^{'}} .

It can be used to perform time reversal on angular momenta.

Other properties

\sum_{m} (- 1)^{j - m} (\begin{matrix} j & j & J \\ m & - m & 0 \end{matrix}) = \sqrt{2 j + 1} δ_{J, 0} .

\frac{1}{2} \int_{- 1}^{1} P_{l_{1}} (x) P_{l_{2}} (x) P_{l} (x) d x = {(\begin{matrix} l & l_{1} & l_{2} \\ 0 & 0 & 0 \end{matrix})}^{2},

where P are Legendre polynomials.

Relation to Racah $V$ -coefficients

Wigner 3-j symbols are related to Racah $V$ -coefficients^[6] by a simple phase:

V (j_{1} j_{2} j_{3}; m_{1} m_{2} m_{3}) = (- 1)^{j_{1} - j_{2} - j_{3}} (\begin{matrix} j_{1} & j_{2} & j_{3} \\ m_{1} & m_{2} & m_{3} \end{matrix}) .

References

↑ ^1.0 ^1.1 Wigner, E. P. (1951). "On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups". In Wightman, Arthur S. (ed.). The Collected Works of Eugene Paul Wigner. 3. pp. 608–654. doi:10.1007/978-3-662-02781-3_42. ISBN 978-3-642-08154-5.
↑ Regge, T. (1958). "Symmetry Properties of Clebsch-Gordan Coefficients". Nuovo Cimento. 10 (3): 544. Bibcode:1958NCim...10..544R. doi:10.1007/BF02859841. S2CID 122299161.
↑ Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.
↑ Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.
↑ P. E. S. Wormer; J. Paldus (2006). "Angular Momentum Diagrams". Advances in Quantum Chemistry. Elsevier. 51: 59–124. Bibcode:2006AdQC...51...59W. doi:10.1016/S0065-3276(06)51002-0. ISBN 9780120348510. ISSN 0065-3276.
↑ Racah, G. (1942). "Theory of Complex Spectra II". Physical Review. 62 (9–10): 438–462. Bibcode:1942PhRv...62..438R. doi:10.1103/PhysRev.62.438.

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Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.

External links

Stone, Anthony. "Wigner coefficient calculator".
Volya, A. "Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator". Archived from the original on 2007-09-29. (Numerical)
Stevenson, Paul (2002). "Clebsch-O-Matic". Computer Physics Communications. 147 (3): 853–858. Bibcode:2002CoPhC.147..853S. doi:10.1016/S0010-4655(02)00462-9.
369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (Numerical)
Frederik J Simons: Matlab software archive, the code THREEJ.M
Sage (mathematics software) Gives exact answer for any value of j, m
Johansson, H. T.; Forssén, C. "(WIGXJPF)". (accurate; C, fortran, python)
Johansson, H. T. "(FASTWIGXJ)". (fast lookup, accurate; C, fortran)

[Wigner1951-1] 1.0 ^1.1 Wigner, E. P. (1951). "On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups". In Wightman, Arthur S. (ed.). The Collected Works of Eugene Paul Wigner. 3. pp. 608–654. doi:10.1007/978-3-662-02781-3_42. ISBN 978-3-642-08154-5.

[2] Regge, T. (1958). "Symmetry Properties of Clebsch-Gordan Coefficients". Nuovo Cimento. 10 (3): 544. Bibcode:1958NCim...10..544R. doi:10.1007/BF02859841. S2CID 122299161.

[3] Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.

[4] Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.

[WormerPaldus2006-5] P. E. S. Wormer; J. Paldus (2006). "Angular Momentum Diagrams". Advances in Quantum Chemistry. Elsevier. 51: 59–124. Bibcode:2006AdQC...51...59W. doi:10.1016/S0065-3276(06)51002-0. ISBN 9780120348510. ISSN 0065-3276.

[6] Racah, G. (1942). "Theory of Complex Spectra II". Physical Review. 62 (9–10): 438–462. Bibcode:1942PhRv...62..438R. doi:10.1103/PhysRev.62.438.

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3-j symbol

Contents

Mathematical relation to Clebsch–Gordan coefficients

Definitional relation to Clebsch–Gordan coefficients

Selection rules

Symmetry properties

Orthogonality relations

Relation to spherical harmonics

Relation to integrals of spin-weighted spherical harmonics

Recursion relations

Asymptotic expressions

Metric tensor

Other properties

Relation to Racah $V$ -coefficients

See also

References

External links

Navigation menu

3-j symbol

Mathematical relation to Clebsch–Gordan coefficients

Definitional relation to Clebsch–Gordan coefficients

Selection rules

Symmetry properties

Orthogonality relations

Relation to spherical harmonics

Relation to integrals of spin-weighted spherical harmonics

Recursion relations

Asymptotic expressions

Metric tensor

Other properties

Relation to Racah V-coefficients

See also

References

External links

Navigation menu

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Relation to Racah $V$ -coefficients