Clebsch–Gordan coefficients

Template:Short description Template:Use American EnglishIn 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 (i.e., a reducible representation 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.

Review of the angular momentum operators

Angular momentum operators are self-adjoint operators Template:Math, Template:Math, and Template:Math that satisfy the commutation relations $\begin{matrix} [j_{k}, j_{l}] \equiv j_{k} j_{l} - j_{l} j_{k} = i ℏ ε_{k l m} j_{m} & k, l, m & \in {x, y, z}, \end{matrix}$ where Template:Math is the Levi-Civita symbol. Together the three operators define a vector operator, a rank one Cartesian tensor operator, $𝐣 = (j_{x}, j_{y}, j_{z}) .$ It is 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 Template:Math as the inner product of Template:Math with itself: $𝐣^{2} = j_{x}^{2} + j_{y}^{2} + j_{z}^{2} .$ This is an example of a Casimir operator. It is diagonal and its eigenvalue characterizes the particular irreducible representation of the angular momentum algebra $𝔰 𝔬 (3, ℝ) ≅ 𝔰 𝔲 (2)$ . 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 (Template:Math) and lowering (Template:Math) operators, the so-called ladder operators, $j_{\pm} = j_{x} \pm i j_{y} .$

Spherical basis for angular momentum eigenstates

It can be shown from the above definitions that Template:Math commutes with Template:Math, Template:Math, and Template:Math: $\begin{matrix} [𝐣^{2}, j_{k}] = 0 & k & \in {x, y, z} . \end{matrix}$

When two Hermitian operators commute, a common set of eigenstates exists. Conventionally, Template:Math and Template:Math are chosen. From the commutation relations, the possible eigenvalues can be found. These eigenstates are denoted Template:Math where Template:Math is the angular momentum quantum number and Template:Math is the angular momentum projection onto the z-axis.

They comprise the spherical basis, are complete, and satisfy the following eigenvalue equations, $\begin{matrix} 𝐣^{2} | j m ⟩ & = ℏ^{2} j (j + 1) | j m ⟩, & j & \in {0, \frac{1}{2}, 1, \frac{3}{2}, \dots} \\ j_{z} | j m ⟩ & = ℏ m | j m ⟩, & m & \in {- j, - j + 1, \dots, j} . \end{matrix}$

The raising and lowering operators can be used to alter the value of Template:Math, $j_{\pm} | j m ⟩ = ℏ C_{\pm} (j, m) | j (m \pm 1) ⟩,$ where the ladder coefficient is given by: Template:NumBlk

In principle, one may also introduce a (possibly complex) phase factor in the definition of $C_{\pm} (j, m)$ . 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, $⟨ j m | j^{'} m^{'} ⟩ = δ_{j, j^{'}} δ_{m, m^{'}} .$

Here the italicized Template:Math and Template:Math denote integer or half-integer angular momentum quantum numbers of a particle or of a system. On the other hand, the roman Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, and Template:Math denote operators. The $δ$ symbols are Kronecker deltas.

Template:See also

Tensor product space

We now consider systems with two physically different angular momenta Template:Math and Template:Math. 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 $V_{1}$ of dimension $2 j_{1} + 1$ and also on a space $V_{2}$ of dimension $2 j_{2} + 1$ . We are then going to define a family of "total angular momentum" operators acting on the tensor product space $V_{1} \otimes V_{2}$ , which has dimension $(2 j_{1} + 1) (2 j_{2} + 1)$ . The action of the total angular momentum operator on this space constitutes a representation of the SU(2) Lie algebra, but a reducible one. The reduction of this reducible representation into irreducible pieces is the goal of Clebsch–Gordan theory.

Let Template:Math be the Template:Math-dimensional vector space spanned by the states $\begin{matrix} | j_{1} m_{1} ⟩, & m_{1} & \in {- j_{1}, - j_{1} + 1, \dots, j_{1}} \end{matrix},$ and Template:Math the Template:Math-dimensional vector space spanned by the states $\begin{matrix} | j_{2} m_{2} ⟩, & m_{2} & \in {- j_{2}, - j_{2} + 1, \dots, j_{2}} \end{matrix} .$

The tensor product of these spaces, Template:Math, has a Template:Math-dimensional uncoupled basis $| j_{1} m_{1} j_{2} m_{2} ⟩ \equiv | j_{1} m_{1} ⟩ \otimes | j_{2} m_{2} ⟩, m_{1} \in {- j_{1}, - j_{1} + 1, \dots, j_{1}}, m_{2} \in {- j_{2}, - j_{2} + 1, \dots, j_{2}} .$ Angular momentum operators are defined to act on states in Template:Math in the following manner: $(𝐣 \otimes 1) | j_{1} m_{1} j_{2} m_{2} ⟩ \equiv 𝐣 | j_{1} m_{1} ⟩ \otimes | j_{2} m_{2} ⟩$ and $(1 \otimes 𝐣) | j_{1} m_{1} j_{2} m_{2} ⟩ \equiv | j_{1} m_{1} ⟩ \otimes 𝐣 | j_{2} m_{2} ⟩,$ where Template:Math denotes the identity operator.

The total^{[nb 1]} angular momentum operators are defined by the coproduct (or tensor product) of the two representations acting on Template:Math,

$𝐉 \equiv 𝐣_{1} \otimes 1 + 1 \otimes 𝐣_{2} .$

The total angular momentum operators can be shown to satisfy the very same commutation relations, $[J_{k}, J_{l}] = i ℏ ε_{k l m} J_{m},$ where Template:Math. Indeed, the preceding construction is the standard method^[4] 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, $\begin{matrix} 𝐉^{2} | [j_{1} j_{2}] J M ⟩ & = ℏ^{2} J (J + 1) | [j_{1} j_{2}] J M ⟩ \\ J_{z} | [j_{1} j_{2}] J M ⟩ & = ℏ M | [j_{1} j_{2}] J M ⟩ \end{matrix}$ for Template:Math. Note that it is common to omit the Template:Math part.

The total angular momentum quantum number Template:Math must satisfy the triangular condition that $| j_{1} - j_{2} | \leq J \leq j_{1} + j_{2},$ such that the three nonnegative integer or half-integer values could correspond to the three sides of a triangle.^[5]

The total number of total angular momentum eigenstates is necessarily equal to the dimension of Template:Math: $\sum_{J = | j_{1} - j_{2} |}^{j_{1} + j_{2}} (2 J + 1) = (2 j_{1} + 1) (2 j_{2} + 1) .$ As this computation suggests, the tensor product representation decomposes as the direct sum of one copy of each of the irreducible representations of dimension $2 J + 1$ , where $J$ ranges from $| j_{1} - j_{2} |$ to $j_{1} + j_{2}$ in increments of 1.^[6] As an example, consider the tensor product of the three-dimensional representation corresponding to $j_{1} = 1$ with the two-dimensional representation with $j_{2} = 1 / 2$ . The possible values of $J$ are then $J = 1 / 2$ and $J = 3 / 2$ . 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 Template:Math: $⟨ J M | J^{'} M^{'} ⟩ = δ_{J, J^{'}} δ_{M, M^{'}} .$

These rules may be iterated to, e.g., combine Template:Mvar doublets (Template:Mvar=1/2) to obtain the Clebsch-Gordan decomposition series, (Catalan's triangle), $𝟐^{\otimes n} = ⨁_{k = 0}^{⌊ n / 2 ⌋} (\frac{n + 1 - 2 k}{n + 1} (\binom{n + 1}{k})) (𝐧 + 𝟏 - 𝟐 𝐤),$ where $⌊ n / 2 ⌋$ is the integer floor function; and the number preceding the boldface irreducible representation dimensionality (Template:Math) label indicates multiplicity of that representation in the representation reduction.^[7] For instance, from this formula, addition of three spin 1/2s yields a spin 3/2 and two spin 1/2s, $𝟐 \otimes 𝟐 \otimes 𝟐 = 𝟒 \oplus 𝟐 \oplus 𝟐$ .

Formal definition of Clebsch–Gordan coefficients

The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis

Template:NumBlk

The expansion coefficients

$⟨ j_{1} m_{1} j_{2} m_{2} | J M ⟩$

are the Clebsch–Gordan coefficients. Note that some authors write them in a different order such as Template:Math. Another common notation is Template:Math.

Applying the operators

$\begin{matrix} J & = j \otimes 1 + 1 \otimes j \\ J_{z} & = j_{z} \otimes 1 + 1 \otimes j_{z} \end{matrix}$

to both sides of the defining equation shows that the Clebsch–Gordan coefficients can only be nonzero when

$\begin{matrix} | j_{1} - j_{2} | \leq J & \leq j_{1} + j_{2} \\ M & = m_{1} + m_{2} . \end{matrix}$

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 $J_{\pm} = j_{\pm} \otimes 1 + 1 \otimes j_{\pm}$ to the left hand side of the defining equation gives $\begin{matrix} J_{\pm} | [j_{1} j_{2}] J M ⟩ & = ℏ C_{\pm} (J, M) | [j_{1} j_{2}] J (M \pm 1) ⟩ \\ = ℏ C_{\pm} (J, M) \sum_{m_{1}, m_{2}} | j_{1} m_{1} j_{2} m_{2} ⟩ ⟨ j_{1} m_{1} j_{2} m_{2} | J (M \pm 1) ⟩ \end{matrix}$ Applying the same operators to the right hand side gives $\begin{matrix} J_{\pm} & \sum_{m_{1}, m_{2}} | j_{1} m_{1} j_{2} m_{2} ⟩ ⟨ j_{1} m_{1} j_{2} m_{2} | J M ⟩ \\ = ℏ & \sum_{m_{1}, m_{2}} (C_{\pm} (j_{1}, m_{1}) | j_{1} (m_{1} \pm 1) j_{2} m_{2} ⟩ + C_{\pm} (j_{2}, m_{2}) | j_{1} m_{1} j_{2} (m_{2} \pm 1) ⟩) ⟨ j_{1} m_{1} j_{2} m_{2} | J M ⟩ \\ = ℏ & \sum_{m_{1}, m_{2}} | j_{1} m_{1} j_{2} m_{2} ⟩ (C_{\pm} (j_{1}, m_{1} \mp 1) ⟨ j_{1} (m_{1} \mp 1) j_{2} m_{2} | J M ⟩ + C_{\pm} (j_{2}, m_{2} \mp 1) ⟨ j_{1} m_{1} j_{2} (m_{2} \mp 1) | J M ⟩) . \end{matrix}$

Combining these results gives recursion relations for the Clebsch–Gordan coefficients, where Template:Math was defined in Template:EquationNote: $C_{\pm} (J, M) ⟨ j_{1} m_{1} j_{2} m_{2} | J (M \pm 1) ⟩ = C_{\pm} (j_{1}, m_{1} \mp 1) ⟨ j_{1} (m_{1} \mp 1) j_{2} m_{2} | J M ⟩ + C_{\pm} (j_{2}, m_{2} \mp 1) ⟨ j_{1} m_{1} j_{2} (m_{2} \mp 1) | J M ⟩ .$

Taking the upper sign with the condition that Template:Math gives initial recursion relation: $0 = C_{+} (j_{1}, m_{1} - 1) ⟨ j_{1} (m_{1} - 1) j_{2} m_{2} | J J ⟩ + C_{+} (j_{2}, m_{2} - 1) ⟨ j_{1} m_{1} j_{2} (m_{2} - 1) | J J ⟩ .$ In the Condon–Shortley phase convention, one adds the constraint that

⟨ j_{1} j_{1} j_{2} (J - j_{1}) | J J ⟩ > 0

(and is therefore also real). The Clebsch–Gordan coefficients Template:Math 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 Template:Math must be one.

The lower sign in the recursion relation can be used to find all the Clebsch–Gordan coefficients with Template:Math. 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

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Orthogonality relations

These are most clearly written down by introducing the alternative notation $⟨ J M | j_{1} m_{1} j_{2} m_{2} ⟩ \equiv ⟨ j_{1} m_{1} j_{2} m_{2} | J M ⟩$

The first orthogonality relation is $\sum_{J = | j_{1} - j_{2} |}^{j_{1} + j_{2}} \sum_{M = - J}^{J} ⟨ j_{1} m_{1} j_{2} m_{2} | J M ⟩ ⟨ J M | j_{1} {m_{1}}^{'} j_{2} {m_{2}}^{'} ⟩ = ⟨ j_{1} m_{1} j_{2} m_{2} | j_{1} {m_{1}}^{'} j_{2} {m_{2}}^{'} ⟩ = δ_{m_{1}, {m_{1}}^{'}} δ_{m_{2}, {m_{2}}^{'}}$ (derived from the fact that $𝟏 = \sum_{x} | x ⟩ ⟨ x |$ ) and the second one is $\sum_{m_{1}, m_{2}} ⟨ J M | j_{1} m_{1} j_{2} m_{2} ⟩ ⟨ j_{1} m_{1} j_{2} m_{2} | J^{'} M^{'} ⟩ = ⟨ J M | J^{'} M^{'} ⟩ = δ_{J, J^{'}} δ_{M, M^{'}} .$

Special cases

For Template:Math the Clebsch–Gordan coefficients are given by $⟨ j_{1} m_{1} j_{2} m_{2} | 0 0 ⟩ = δ_{j_{1}, j_{2}} δ_{m_{1}, - m_{2}} \frac{(- 1)^{j_{1} - m_{1}}}{\sqrt{2 j_{1} + 1}} .$

For Template:Math and Template:Math we have $⟨ j_{1} j_{1} j_{2} j_{2} | (j_{1} + j_{2}) (j_{1} + j_{2}) ⟩ = 1 .$

For Template:Math and Template:Math we have $⟨ j_{1} m_{1} j_{1} (- m_{1}) | (2 j_{1}) 0 ⟩ = \frac{(2 j_{1})!^{2}}{(j_{1} - m_{1})! (j_{1} + m_{1})! \sqrt{(4 j_{1})!}} .$

For Template:Math we have $⟨ j_{1} j_{1} j_{1} (- j_{1}) | J 0 ⟩ = (2 j_{1})! \sqrt{\frac{2 J + 1}{(J + 2 j_{1} + 1)! (2 j_{1} - J)!}} .$

For Template:Math, Template:Math we have $\begin{matrix} ⟨ j_{1} m 1 0 | (j_{1} + 1) m ⟩ & = \sqrt{\frac{(j_{1} - m + 1) (j_{1} + m + 1)}{(2 j_{1} + 1) (j_{1} + 1)}} \\ ⟨ j_{1} m 1 0 | j_{1} m ⟩ & = \frac{m}{\sqrt{j_{1} (j_{1} + 1)}} \\ ⟨ j_{1} m 1 0 | (j_{1} - 1) m ⟩ & = - \sqrt{\frac{(j_{1} - m) (j_{1} + m)}{j_{1} (2 j_{1} + 1)}} \end{matrix}$

For Template:Math we have $\begin{matrix} ⟨ j_{1} (M - \frac{1}{2}) \frac{1}{2} \frac{1}{2} | (j_{1} \pm \frac{1}{2}) M ⟩ & = \pm \sqrt{\frac{1}{2} (1 \pm \frac{M}{j_{1} + \frac{1}{2}})} \\ ⟨ j_{1} (M + \frac{1}{2}) \frac{1}{2} (- \frac{1}{2}) | (j_{1} \pm \frac{1}{2}) M ⟩ & = \sqrt{\frac{1}{2} (1 \mp \frac{M}{j_{1} + \frac{1}{2}})} \end{matrix}$

Symmetry properties

$\begin{matrix} ⟨ j_{1} m_{1} j_{2} m_{2} | J M ⟩ & = (- 1)^{j_{1} + j_{2} - J} ⟨ j_{1} (- m_{1}) j_{2} (- m_{2}) | J (- M) ⟩ \\ = (- 1)^{j_{1} + j_{2} - J} ⟨ j_{2} m_{2} j_{1} m_{1} | J M ⟩ \\ = (- 1)^{j_{1} - m_{1}} \sqrt{\frac{2 J + 1}{2 j_{2} + 1}} ⟨ j_{1} m_{1} J (- M) | j_{2} (- m_{2}) ⟩ \\ = (- 1)^{j_{2} + m_{2}} \sqrt{\frac{2 J + 1}{2 j_{1} + 1}} ⟨ J (- M) j_{2} m_{2} | j_{1} (- m_{1}) ⟩ \\ = (- 1)^{j_{1} - m_{1}} \sqrt{\frac{2 J + 1}{2 j_{2} + 1}} ⟨ J M j_{1} (- m_{1}) | j_{2} m_{2} ⟩ \\ = (- 1)^{j_{2} + m_{2}} \sqrt{\frac{2 J + 1}{2 j_{1} + 1}} ⟨ j_{2} (- m_{2}) J M | j_{1} m_{1} ⟩ \end{matrix}$

A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3-j symbols using Template:EquationNote. 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 Template:Math is not necessarily Template:Math for a given quantum number Template:Math unless it can be proven to be an integer. Instead, it is replaced by the following weaker rule: $(- 1)^{4 k} = 1$ for any angular-momentum-like quantum number Template:Math.

Nonetheless, a combination of Template:Math and Template:Math is always an integer, so the stronger rule applies for these combinations: $(- 1)^{2 (j_{i} - m_{i})} = 1$ This identity also holds if the sign of either Template:Math or Template:Math or both is reversed.

It is useful to observe that any phase factor for a given Template:Math pair can be reduced to the canonical form: $(- 1)^{a j_{i} + b (j_{i} - m_{i})}$ where Template:Math and Template:Math (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 Template:Math pairs such as the one described in the next paragraph.)

An additional rule holds for combinations of Template:Math, Template:Math, and Template:Math that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol: $(- 1)^{2 (j_{1} + j_{2} + j_{3})} = 1$ This identity also holds if the sign of any Template:Math is reversed, or if any of them are substituted with an Template:Math instead.

Template:Anchor Relation to Wigner 3-j symbols

Clebsch–Gordan coefficients are related to Wigner 3-j symbols which have more convenient symmetry relations. Template:NumBlk

The factor Template:Math is due to the Condon–Shortley constraint that Template:Math, while Template:Math is due to the time-reversed nature of Template:Math.

This allows to reach the general expression:

\begin{matrix} ⟨ j_{1} m_{1} j_{2} m_{2} | J M ⟩ & \equiv δ (m_{1} + m_{2}, M) \sqrt{\frac{(2 J + 1) (j_{1} + j_{2} - J)! (j_{1} - j_{2} + J)! (- j_{1} + j_{2} + J)!}{(j_{1} + j_{2} + J + 1)!}} \times \\ \times \sqrt{(j_{1} - m_{1})! (j_{1} + m_{1})! (j_{2} - m_{2})! (j_{2} + m_{2})! (J - M)! (J + M)!} \times \\ \times \sum_{k = K}^{N} \frac{(- 1)^{k}}{k! (j_{1} + j_{2} - J - k)! (j_{1} - m_{1} - k)! (j_{2} + m_{2} - k)! (J - j_{2} + m_{1} + k)! (J - j_{1} - m_{2} + k)!}, \end{matrix} .

The summation is performed over those integer values Template:Mvar for which the argument of each factorial in the denominator is non-negative, i.e. summation limits Template:Mvar and Template:Mvar are taken equal: the lower one $K = \max (0, j_{2} - J - m_{1}, j_{1} - J + m_{2}),$ the upper one $N = \min (j_{1} + j_{2} - J, j_{1} - m_{1}, j_{2} + m_{2}) .$ Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example, $j_{3} > j_{1} + j_{2}$ or $j_{1} < m_{1}$ are automatically set to zero.

Relation to Wigner D-matrices

Template:Main $\begin{matrix} \int_{0}^{2 π} d α \int_{0}^{π} \sin β d β \int_{0}^{2 π} d γ D_{M, K}^{J} (α, β, γ)^{*} D_{m_{1}, k_{1}}^{j_{1}} (α, β, γ) D_{m_{2}, k_{2}}^{j_{2}} (α, β, γ) \\ = & \frac{8 π^{2}}{2 J + 1} ⟨ j_{1} m_{1} j_{2} m_{2} | J M ⟩ ⟨ j_{1} k_{1} j_{2} k_{2} | J K ⟩ \end{matrix}$

Relation to spherical harmonics

In the case where integers are involved, the coefficients can be related to integrals of spherical harmonics: $\int_{4 π} Y_{ℓ_{1}}^{m_{1}}^{*} (Ω) Y_{ℓ_{2}}^{m_{2}}^{*} (Ω) Y_{L}^{M} (Ω) d Ω = \sqrt{\frac{(2 ℓ_{1} + 1) (2 ℓ_{2} + 1)}{4 π (2 L + 1)}} ⟨ ℓ_{1} 0 ℓ_{2} 0 | L 0 ⟩ ⟨ ℓ_{1} m_{1} ℓ_{2} m_{2} | L M ⟩$

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: $Y_{ℓ_{1}}^{m_{1}} (Ω) Y_{ℓ_{2}}^{m_{2}} (Ω) = \sum_{L, M} \sqrt{\frac{(2 ℓ_{1} + 1) (2 ℓ_{2} + 1)}{4 π (2 L + 1)}} ⟨ ℓ_{1} 0 ℓ_{2} 0 | L 0 ⟩ ⟨ ℓ_{1} m_{1} ℓ_{2} m_{2} | L M ⟩ Y_{L}^{M} (Ω)$

Other properties

$\sum_{m} (- 1)^{j - m} ⟨ j m j (- m) | J 0 ⟩ = δ_{J, 0} \sqrt{2 j + 1}$