Spherical basis

In mathematics and physics, a spherical basis is a set of orthonormal basis vectors in three-dimensional Euclidean space that transform irreducibly under rotations of the SO(3) group, making them particularly suited for describing systems with rotational symmetry.^[1] For rank-one tensors (corresponding to vectors), the standard spherical basis vectors are defined as complex combinations of the Cartesian unit vectors \hat{x}, \hat{y}, and \hat{z}:

\hat{e}_{+1} = -\frac{\hat{x} + i \hat{y}}{\sqrt{2}}, \quad \hat{e}_{0} = \hat{z}, \quad \hat{e}_{-1} = \frac{\hat{x} - i \hat{y}}{\sqrt{2}}.

This convention incorporates the Condon-Shortley phase to align with the transformation properties of spherical harmonics Y_1^q(\theta, \phi).^[1] These vectors satisfy the orthonormality condition \hat{e}_q^\dagger \cdot \hat{e}_{q'} = \delta_{q q'}, where the dagger denotes complex conjugation, ensuring they form a complete basis for vector spaces.^[2] The spherical basis is distinguished by its transformation behavior under rotations: a rotation R acts on the basis vectors as R \hat{e}_q = \sum_{q'} D^1_{q' q}(R) \hat{e}_{q'}, where D^1_{q' q}(R) are the Wigner D-matrix elements for angular momentum quantum number j=1.^[1] This mirrors the behavior of angular momentum eigenstates |j, m\rangle, with q playing the role of the magnetic quantum number m (ranging from -1 to +1). In infinitesimal form, the basis satisfies commutation relations with the angular momentum operators J_i, such as [J_z, \hat{e}_q] = q \hbar \hat{e}_q and [J_\pm, \hat{e}_q] = \hbar \sqrt{1(1+1) - q(q \pm 1)} \hat{e}_{q \pm 1}, confirming their equivalence to a spin-1 representation. In quantum mechanics, the spherical basis is fundamental for constructing spherical tensors of arbitrary rank k, which are sets of $2k+1 components T_q^k that transform under rotations according to the Wigner D-matrices.^[3] This framework enables the Wigner-Eckart theorem, which simplifies calculations of matrix elements of tensor operators between angular momentum states by factoring out geometric factors. Applications include atomic and molecular spectroscopy, where dipole operators (rank-1 tensors) couple to selection rules, and nuclear physics for multipole expansions of potentials. Beyond quantum mechanics, spherical bases appear in classical electromagnetism for expanding fields in vector spherical harmonics and in computer graphics for efficient rotation-invariant representations related to spherical harmonics.

Fundamentals in Three Dimensions

Unit Vectors and Explicit Construction

The spherical basis in three dimensions consists of three complex unit vectors \mathbf{e}_+, \mathbf{e}_-, \mathbf{e}_0, defined in terms of the standard Cartesian unit vectors \mathbf{e}_x, \mathbf{e}_y, \mathbf{e}_z as follows:

\mathbf{e}_+ = -\frac{1}{\sqrt{2}} (\mathbf{e}_x + i \mathbf{e}_y), \quad \mathbf{e}_- = \frac{1}{\sqrt{2}} (\mathbf{e}_x - i \mathbf{e}_y), \quad \mathbf{e}_0 = \mathbf{e}_z.

These vectors form a complete basis for three-dimensional vectors, where the subscript + corresponds to the magnetic quantum number m = +1, - to m = -1, and $0 to m = 0. Any vector \mathbf{A} can be decomposed into its spherical components as

\mathbf{A} = A_+ \mathbf{e}_+ + A_- \mathbf{e}_- + A_0 \mathbf{e}_0,

with the coefficients A_+, A_-, A_0 given explicitly by the projection onto the conjugate basis vectors:

A_+ = \frac{1}{\sqrt{2}} (-A_x + i A_y), \quad A_- = \frac{1}{\sqrt{2}} (A_x + i A_y), \quad A_0 = A_z,

where A_x = \mathbf{A} \cdot \mathbf{e}_x, A_y = \mathbf{A} \cdot \mathbf{e}_y, and A_z = \mathbf{A} \cdot \mathbf{e}_z are the Cartesian components. Note the sign convention in A_+, which aligns with the choice of phase in \mathbf{e}_+ to ensure consistency with standard angular momentum conventions. This basis is motivated by its utility in quantum mechanics and classical field theory, where the vectors are eigenvectors of the angular momentum operator J_z with eigenvalues + \hbar, 0, -\hbar, respectively, thereby diagonalizing rotations about the z-axis. Additionally, it simplifies the expansion of vector fields in terms of spherical harmonics by transforming as a rank-1 irreducible tensor under rotations.

Relation to Cartesian Coordinates

The spherical basis utilizes complex-valued unit vectors to facilitate the decomposition of vectors in a manner that aligns with the irreducible representations of the rotation group SO(3), particularly simplifying calculations in systems exhibiting rotational symmetry around the z-axis. This framework assumes familiarity with vector spaces over the complex numbers, where the standard inner product incorporates complex conjugation to ensure orthonormality; the complex nature of the basis elegantly captures circular symmetries that are cumbersome in real-valued Cartesian coordinates. The mapping from Cartesian components (A_x, A_y, A_z) of a vector \mathbf{A} to its spherical components (A_+, A_0, A_-) is achieved via a linear transformation represented by the 3×3 matrix U, where the rows correspond to the spherical indices q = +1, 0, -1 (or equivalently q = 1, 0, -1) and the columns to the Cartesian indices x, y, z:

U = \begin{pmatrix} -\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \\ \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \end{pmatrix}.

Thus, the spherical components are obtained as \begin{pmatrix} A_+ \\ A_0 \\ A_- \end{pmatrix} = U \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix}. This matrix embodies the explicit construction of the spherical unit vectors in terms of the Cartesian ones, with \mathbf{e}_+ = -\frac{1}{\sqrt{2}} (\mathbf{e}_x + i \mathbf{e}_y), \mathbf{e}_0 = \mathbf{e}_z, and \mathbf{e}_- = \frac{1}{\sqrt{2}} (\mathbf{e}_x - i \mathbf{e}_y). The inverse transformation, which expresses the Cartesian components in terms of the spherical ones, is given by

A_x = \frac{1}{\sqrt{2}} (A_- - A_+), \quad A_y = -\frac{i}{\sqrt{2}} (A_- + A_+), \quad A_z = A_0.

In matrix form, this corresponds to multiplication by U^{-1}, yielding \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} = U^{-1} \begin{pmatrix} A_+ \\ A_0 \\ A_- \end{pmatrix}. These relations follow directly from solving the system defined by U, ensuring the bidirectional equivalence between the two bases. Geometrically, the vectors \mathbf{e}_+ and \mathbf{e}_- represent right- and left-handed circular polarizations lying in the xy-plane, perpendicular to the polar z-axis along which \mathbf{e}_0 points linearly; this decomposition resolves any vector into components that are eigenstates of rotations about the z-axis, thereby highlighting the intrinsic rotational invariance of physical quantities like angular momentum.

Alternative Definitions

Commutator Approach

The commutator approach defines the spherical basis through the Lie algebra of the rotation group SO(3), realized by the angular momentum operators J_x, J_y, and J_z satisfying the commutation relations [J_x, J_y] = i \hbar J_z and cyclic permutations thereof.^[4] These relations encode the infinitesimal generators of rotations, ensuring the basis transforms irreducibly under the group action. In this framework, the spherical basis vectors e_{+1}, e_0, and e_{-1} emerge as the standard basis for the three-dimensional representation corresponding to angular momentum quantum number j=1. To construct the basis, introduce the raising operator J_+ = J_x + i J_y and lowering operator J_- = J_x - i J_y, which satisfy [J_z, J_\pm] = \pm \hbar J_\pm, [J_+, J_-] = 2 \hbar J_z.^[4] The basis vectors are simultaneous eigenvectors of the Casimir operator J^2 = J_x^2 + J_y^2 + J_z^2 (with eigenvalue \hbar^2 j(j+1) = 2 \hbar^2) and J_z (with eigenvalues \hbar m for m = +1, 0, -1), such that J_z e_m = \hbar m e_m. The highest-weight condition J_+ e_{+1} = 0 annihilates the m=1 state, while the lowest-weight condition J_- e_{-1} = 0 does so for m=-1; for the m=0 state, J_z e_0 = 0. Applying the ladder operators connects the states, with matrix elements like J_+ e_0 = \hbar \sqrt{2} e_{+1} and J_- e_0 = \hbar \sqrt{2} e_{-1}, incorporating the normalization factor \sqrt{2} characteristic of the j=1 irreducible representation.^[5] For a vector \mathbf{A} expressed in the spherical basis as \mathbf{A} = \sum_m A_m e_m, the components A_m behave as rank-1 spherical tensor operators, obeying commutation relations with the angular momentum components J_q (where q = 0, \pm 1): [J_z, A_m] = \hbar m A_m, [J_+, A_m] = \hbar \sqrt{(1 - m)(2 + m)} A_{m+1}, and [J_-, A_m] = \hbar \sqrt{(1 + m)(2 - m)} A_{m-1}.^[5] More generally, [J_q, A_m] = \hbar \sum_{m'} C^{(1)}_{m' m; 1 q} A_{m'}, where C are Clebsch-Gordan coefficients for the j=1 representation, reducing to diagonal terms \hbar m \delta_{q m} A_m when q=0 and shifts otherwise. This derives the basis directly from the j=1 irreducible representation of SO(3), ensuring unitarity and proper transformation properties. This approach offers an intrinsic definition tied to the symmetry group, rendering it independent of any specific coordinate system and highlighting the basis's role in representing vector quantities under rotations.^[4]

Rotation Operator Approach

The rotation operator approach defines the spherical basis through the action of finite rotation operators from the SO(3) group, leveraging the irreducible spin-1 representation to ensure the basis vectors transform covariantly under rotations. Specifically, the basis vectors e_m (with m = +1, 0, -1) satisfy e_m' = \sum_{m'} D^{(1)}_{m' m}(R) e_{m'}, where D^{(1)}_{m' m}(R) are the elements of the Wigner D-matrix for the j=1 representation corresponding to the rotation R.^[1] This formulation guarantees that the three-dimensional vector space spanned by the basis is irreducible under SO(3), meaning no proper subspace is invariant under all rotations.^[1] For rotations around the z-axis by an angle \phi, the D-matrix is diagonal, leading to simple phase transformations: e_{+} \to e^{-i \phi} e_{+}, e_0 \to e_0, and e_{-} \to e^{i \phi} e_{-}.^[1] These phase shifts reflect the azimuthal quantum numbers associated with each basis vector, with e_{+} and e_{-} corresponding to helical components that acquire opposite phases under azimuthal rotation.^[1] In general, the spherical basis can be constructed by taking the columns of the inverse D-matrix for a reference rotation, which aligns the basis with the standard form of the representation and ensures irreducibility under arbitrary elements of SO(3).^[1] This approach contrasts with the real spherical unit vectors \hat{e}_r, \hat{e}_\theta, \hat{e}_\phi, which depend on position and do not form a fixed global basis for vector transformations; the complex spherical basis, being fixed and independent of position, better diagonalizes rotation operators, particularly around the z-axis, facilitating analysis in rotationally symmetric systems.^[1]

Key Properties

Orthonormality and Unitarity

The spherical basis vectors \mathbf{e}_m for m = +1, 0, -1 form an orthonormal set under the complex inner product defined as \langle \mathbf{a} | \mathbf{b} \rangle = \mathbf{a}^\dagger \cdot \mathbf{b} = \sum_{i=1}^3 a_i^* b_i, satisfying \langle \mathbf{e}_m | \mathbf{e}_{m'} \rangle = \delta_{m m'}.^[1] For instance, \langle \mathbf{e}_{+1} | \mathbf{e}_{+1} \rangle = 1 and \langle \mathbf{e}_{+1} | \mathbf{e}_{-1} \rangle = 0.^[6] To verify this, substitute the explicit expressions in the Cartesian basis: \mathbf{e}_{+1} = -\frac{\hat{\mathbf{x}} + i \hat{\mathbf{y}}}{\sqrt{2}}, \mathbf{e}_{-1} = \frac{\hat{\mathbf{x}} - i \hat{\mathbf{y}}}{\sqrt{2}}, and \mathbf{e}_0 = \hat{\mathbf{z}}.^[1] The Cartesian unit vectors \hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}} are real and orthonormal, so compute the inner products componentwise. For the norm of \mathbf{e}_{+1},

\langle \mathbf{e}_{+1} | \mathbf{e}_{+1} \rangle = \left( -\frac{1}{\sqrt{2}} \right)^* \left( -\frac{1}{\sqrt{2}} \right) + \left( -i \frac{1}{\sqrt{2}} \right)^* \left( -i \frac{1}{\sqrt{2}} \right) + 0 = \frac{1}{2} + \frac{1}{2} = [1](/page/1),

where the y-component simplifies as (i / \sqrt{2})(-i / \sqrt{2}) = 1/2. Similarly, \langle \mathbf{e}_0 | \mathbf{e}_0 \rangle = [1](/page/1) follows directly, and \langle \mathbf{e}_{+1} | \mathbf{e}_0 \rangle = 0 since the z-components vanish for \mathbf{e}_{+1}. For the off-diagonal \langle \mathbf{e}_{+1} | \mathbf{e}_{-1} \rangle,

\langle \mathbf{e}_{+1} | \mathbf{e}_{-1} \rangle = \left( -\frac{1}{\sqrt{2}} \right)^* \left( \frac{1}{\sqrt{2}} \right) + \left( -i \frac{1}{\sqrt{2}} \right)^* \left( -i \frac{1}{\sqrt{2}} \right) = -\frac{1}{2} + \frac{1}{2} = 0,

with the imaginary contributions from the y-components canceling against the real x-contributions; analogous calculations hold for other pairs.^[6] The transformation matrix U mapping Cartesian components to spherical components (or vice versa) is unitary, satisfying U^\dagger U = I. The columns of U are the Cartesian components of the \mathbf{e}_m, so orthonormality of the columns implies U^\dagger U = I, as the inner products of distinct columns are zero and norms are unity. Explicitly, the matrix elements follow from the basis expressions above, and row/column inner products match the \delta_{m m'}.^[1]

U = \begin{pmatrix} -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ -\frac{i}{\sqrt{2}} & 0 & -\frac{i}{\sqrt{2}} \\ 0 & 1 & 0 \end{pmatrix},

This unitarity ensures preservation of vector norms and angles when expressed in spherical components, such that \sum_m |V_m|^2 = \sum_c |V_c|^2 for any vector \mathbf{V}, which is essential for Parseval's theorem in multipole expansions and rotational invariants.^[6]

Basis Transformation Matrices

The transformation between the Cartesian basis \{\hat{e}_x, \hat{e}_y, \hat{e}_z\} and the spherical basis \{\hat{e}_{+1}, \hat{e}_0, \hat{e}_{-1}\} is given by a unitary 3×3 matrix U, where the columns correspond to the components of the spherical basis vectors in the Cartesian frame. The explicit elements are:

U = \begin{pmatrix} -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ -\frac{i}{\sqrt{2}} & 0 & -\frac{i}{\sqrt{2}} \\ 0 & 1 & 0 \end{pmatrix},

with rows indexed by x, y, z and columns by +1, 0, -1. Thus, the spherical components of a vector \mathbf{A} are obtained as A_{+1} = -\frac{1}{\sqrt{2}} (A_x + i A_y), A_0 = A_z, and A_{-1} = \frac{1}{\sqrt{2}} (A_x - i A_y). The inverse transformation is U^{-1} = U^\dagger, yielding the Cartesian components from the spherical ones, such as A_x = \frac{1}{\sqrt{2}} (A_{-1} - A_{+1}) and A_y = \frac{i}{\sqrt{2}} (A_{-1} + A_{+1}).^[7] Under a general rotation R parameterized by Euler angles (\alpha, \beta, \gamma), the components in the spherical basis transform via the Wigner D-matrix for l=1: R = D^{(1)}(\alpha, \beta, \gamma), a unitary 3×3 matrix with elements D^{(1)}_{m m'}(\alpha, \beta, \gamma) = e^{-i \alpha m} d^{(1)}_{m m'}(\beta) e^{-i \gamma m'}. The reduced d-matrix elements include d^{(1)}_{1,1}(\beta) = \frac{1 + \cos \beta}{2}, d^{(1)}_{1,0}(\beta) = -\frac{\sin \beta}{\sqrt{2}}, and d^{(1)}_{0,0}(\beta) = \cos \beta, among others derived from the general angular momentum rotation formulas. The transformed components are then A'_m = \sum_{m' = -1}^{1} D^{(1)}_{m m'}(R) A_{m'}.^[1]^[8] For a numerical example, consider a 90° (\beta = \pi/2) rotation around the y-axis (\alpha = \gamma = 0), so \cos \beta = 0 and \sin \beta = 1. The d^{(1)}-matrix simplifies to

d^{(1)}(\pi/2) = \begin{pmatrix} \frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2} \\ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ \frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \end{pmatrix},

with rows and columns indexed by m, m' = 1, 0, -1. Take an initial vector along the x-axis in the spherical basis: A_{+1} = -1/\sqrt{2}, A_0 = 0, A_{-1} = 1/\sqrt{2}. After rotation, the new components are A'_1 = \frac{1}{2} (-1/\sqrt{2}) - \frac{1}{\sqrt{2}} (0) + \frac{1}{2} (1/\sqrt{2}) = 0, A'_0 = \frac{1}{\sqrt{2}} (-1/\sqrt{2}) + 0 + (-\frac{1}{\sqrt{2}}) (1/\sqrt{2}) = - \frac{1}{2} - \frac{1}{2} = -1, and A'_{-1} = \frac{1}{2} (-1/\sqrt{2}) + \frac{1}{\sqrt{2}} (0) + \frac{1}{2} (1/\sqrt{2}) = 0. This corresponds to the vector -\hat{z}, confirming the transformation properties, with the negative sign representing a phase factor of e^{i\pi}.^[8]

Vector Operations

In the spherical basis, the inner product (dot product) of two vectors \mathbf{A} and \mathbf{B} is defined as \mathbf{A} \cdot \mathbf{B} = \sum_{m=-1}^{1} A_m^* B_m, where the complex conjugate on A_m accounts for the non-Hermitian nature of the basis vectors \mathbf{e}_m. This form preserves the invariance of the inner product under rotations, as the transformation matrix relating spherical to Cartesian components is unitary.^[1] The cross product \mathbf{A} \times \mathbf{B} in the spherical basis can be expressed component-wise using the Levi-Civita symbol adapted to the complex basis, yielding the explicit formula

(\mathbf{A} \times \mathbf{B})_1 = i (A_1 B_0 - A_0 B_1), \quad (\mathbf{A} \times \mathbf{B})_0 = i (A_1 B_{-1} - A_{-1} B_1), \quad (\mathbf{A} \times \mathbf{B})_{-1} = i (A_0 B_{-1} - A_{-1} B_0).

This component form arises from the antisymmetric coupling of two rank-1 spherical tensors and ensures the result transforms as a vector under rotations. Applying this to the basis vectors themselves produces the cyclic rules

\mathbf{e}_+ \times \mathbf{e}_- = i \mathbf{e}_0, \quad \mathbf{e}_- \times \mathbf{e}_0 = i \mathbf{e}_+, \quad \mathbf{e}_0 \times \mathbf{e}_+ = i \mathbf{e}_-,

where the imaginary unit i reflects the complex structure of the basis (with \mathbf{e}_+ = \mathbf{e}_1, \mathbf{e}_- = \mathbf{e}_{-1}, and \mathbf{e}_0 = \mathbf{e}_0). These relations follow directly from substituting the basis components into the general formula above. More generally, the cross product components can be written using Clebsch-Gordan coefficients for the coupling of two angular momenta l=1:

(A \times B)_m = \sum_{m_1 m_2} C^{(1)}_{m_1 m_2; m} A_{m_1} B_{m_2},

where the C^{(1)} are the appropriate coefficients for the antisymmetric rank-1 tensor product (up to a conventional phase factor of i or -i in some normalizations to match the Cartesian cross product). This expression leverages the rotational invariance of the coupling scheme. (Varshalovich et al., standard reference for CG coefficients in tensor products) To verify these rules, consider the transformation to Cartesian coordinates, where the standard cross product is well-defined. The spherical components relate to Cartesian via the unitary matrix

\begin{pmatrix} V_x \\ V_y \\ V_z \end{pmatrix} = \frac{1}{\sqrt{2}} \begin{pmatrix} -1 & 0 & 1 \\ i & 0 & i \\ 0 & \sqrt{2} & 0 \end{pmatrix} \begin{pmatrix} V_1 \\ V_0 \\ V_{-1} \end{pmatrix},

with the inverse transforming back. For the basis vectors, \mathbf{e}_1 = -\frac{1}{\sqrt{2}} (\hat{x} + i \hat{y}), \mathbf{e}_{-1} = \frac{1}{\sqrt{2}} (\hat{x} - i \hat{y}), \mathbf{e}_0 = \hat{z}. Computing \mathbf{e}_1 \times \mathbf{e}_{-1} in Cartesian yields i \hat{z} = i \mathbf{e}_0, confirming the rule; similar expansions verify the cyclic permutations.

Extensions to Spherical Tensors

Rank-One Tensors

In the context of spherical bases, rank-one spherical tensors represent the natural extension of vector components into the framework of irreducible representations under rotations. A rank-one spherical tensor T^{(1)}_m, with m = -[1](/page/1), 0, [1](/page/1), consists of three components that transform under rotations according to the representation matrices D^{(1)}_{m' m} of the rotation group SO(3), where these matrices are the Wigner D-functions for angular momentum j = 1.^[9]^[7] This transformation ensures that the tensor components mix among themselves in a manner identical to the spherical basis vectors, establishing their equivalence to the components A_m of a vector in the spherical basis.^[10] The connection to Cartesian tensors is made explicit through a linear transformation that converts the standard Cartesian components T_i (for i = x, y, z) into the spherical form. For a vector treated as a rank-one tensor, the components are given by:

\begin{align*} T^{(1)}_0 &= T_z, \\ T^{(1)}_{1} &= -\frac{1}{\sqrt{2}} (T_x + i T_y), \\ T^{(1)}_{-1} &= \frac{1}{\sqrt{2}} (T_x - i T_y). \end{align*}

This relation simplifies the mixed tensor case to pure vector components, highlighting how the spherical basis diagonalizes rotational transformations while preserving the vector's magnitude and direction.^[10]^[7] The inverse transformation recovers the Cartesian components, such as T_x = \frac{1}{\sqrt{2}} (T^{(1)}_{-1} - T^{(1)}_1 ), ensuring equivalence between the two representations.^[7] The three components of a rank-one spherical tensor form an irreducible representation of SO(3), meaning there are no non-trivial invariant subspaces under group actions; the entire triplet transforms as a fundamental unit corresponding to the three-dimensional vector space.^[9]^[7] This irreducibility distinguishes spherical tensors from their Cartesian counterparts, which may contain reducible parts under rotations. A concrete example is the position vector \mathbf{r}, which serves as a rank-one spherical tensor with components r_m defined analogously to the above transformation from its Cartesian coordinates (x, y, z). These components r_1 = -\frac{1}{\sqrt{2}} (x + i y), r_0 = z, and r_{-1} = \frac{1}{\sqrt{2}} (x - i y) transform covariantly under rotations, illustrating the practical utility in problems involving rotational symmetry, such as in quantum mechanics or electromagnetism.^[9]^[10]

Higher-Rank Constructions

Higher-rank spherical tensors are constructed by coupling lower-rank tensors through tensor products, decomposing the product space into irreducible representations under the rotation group SO(3). For two rank-one spherical tensors T^{(1)}_{q_1} and U^{(1)}_{q_2}, their tensor product spans ranks k = 0, 1, 2, with the components of the rank-k tensor given by

V^{(k)}_q = \sum_{q_1, q_2} \langle 1 \, 1; q_1 q_2 | k q \rangle T^{(1)}_{q_1} U^{(1)}_{q_2},

where \langle 1 \, 1; q_1 q_2 | k q \rangle are the Clebsch-Gordan coefficients ensuring the projection onto the irreducible subspace of total angular momentum k, and q = q_1 + q_2. This coupling isolates the pure spin-k representation from the nine-dimensional product space, with the scalar (k=0), vector (k=1), and quadrupole (k=2) parts corresponding to the trace, antisymmetric, and traceless symmetric portions, respectively.^[11] The components within a fixed rank k can be generated using angular momentum ladder operators J_\pm = J_x \pm i J_y, which act to raise or lower the magnetic quantum number m while preserving the rank. Specifically, the commutation relations are

[J_\pm, T^{(k)}_m] = \hbar \sqrt{k(k+1) - m(m \pm 1)} \, T^{(k)}_{m \pm 1},

allowing construction of T^{(k)}_{m+1} from T^{(k)}_m via T^{(k)}_{m+1} \propto [J_+, T^{(k)}_m]_- / \hbar \sqrt{k(k+1) - m(m+1)}, where the subscript denotes the commutator. This method ensures the tensor transforms irreducibly under rotations, analogous to the action on angular momentum states.^[12] An explicit example is the rank-2 quadrupole tensor Q^{(2)}_m, derived from a symmetric traceless Cartesian tensor Q_{ij} with five independent components. The spherical components are

Q^{(2)}_{\pm 2} = \frac{1}{2} (Q_{xx} - Q_{yy} \pm 2i Q_{xy}),

Q^{(2)}_{\pm 1} = -\frac{1}{\sqrt{2}} (Q_{xz} + i Q_{yz}) \quad \text{(for } +1\text{)}, \quad \frac{1}{\sqrt{2}} (Q_{xz} - i Q_{yz}) \quad \text{(for } -1\text{)},

Q^{(2)}_0 = \frac{1}{\sqrt{6}} (2 Q_{zz} - Q_{xx} - Q_{yy}),

with the factors ensuring orthonormal transformation properties. For even ranks like k=2, the underlying Cartesian tensor is symmetric and traceless (Q_{ii} = 0), projecting out scalar contaminants to yield a pure spin-2 representation.^[13]

Irreducible Representations Under Rotation

Spherical tensors of rank k transform under rotations in the special orthogonal group SO(3) according to the law T^{(k)'}_m = \sum_{m'=-k}^{k} D^{(k)}_{m m'}(R) T^{(k)}_{m'}, where D^{(k)}_{m m'}(R) are the elements of the Wigner D-matrix representing the irreducible representation of dimension $2k+1.^[14]^[15] This transformation ensures that the $2k+1 components of the tensor T^{(k)} behave collectively as a single irreducible representation, generalizing the vector case (k=1) to arbitrary integer ranks k \geq 0.^[11] The irreducibility of this representation is characterized by the fact that the $2k+1 components span a (2k+1)-dimensional subspace under the action of SO(3), which cannot be decomposed into smaller invariant subspaces.^[15] For k=0, the tensor reduces to a scalar invariant under rotations, with no nontrivial invariants existing for k > 0.^[14] This property unifies the transformation behavior across ranks, making spherical tensors fundamental for describing physical systems with rotational symmetry, such as angular momentum in quantum mechanics.^[11] The components T^{(k)}_m of a spherical tensor are intimately related to spherical harmonics Y^{(k)}_m(\theta, \phi), serving as coefficients in the expansion of tensor fields on the unit sphere.^[14] Specifically, a scalar function or potential can be decomposed as \sum_{k=0}^{\infty} \sum_{m=-k}^{k} T^{(k)}_m Y^{(k)}_m(\theta, \phi), where the T^{(k)}_m transform identically to the harmonics under rotations.^[15] This connection highlights the spherical basis as a natural framework for harmonic analysis on the sphere. While the spherical basis is primarily defined for three-dimensional rotations in SO(3), it extends naturally to higher dimensions through representations of SO(n) using hyperspherical harmonics.^[16] However, for n > 3, there is no simple complex basis analogous to the m = -k, \dots, k indexing without invoking more intricate hyperspherical coordinates and analogs.^[17]