Spherical basis

 ( Image Processing ) 

In pure mathematics and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express Tensor operator. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions.

While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use .

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In three dimensions A vector A in 3D Euclidean space can be expressed in the familiar Cartesian coordinate system in the standard basis e _x, e _y, e _z, and coordinates A_x, A_y, A_z:

or any other coordinate system with associated basis set of vectors. From this extend the scalars to allow multiplication by complex numbers, so that we are now working in $\backslash mathbb\{C\}^3$ rather than $\backslash mathbb\{R\}^3$.

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Basis definition In the spherical bases denoted e₊, e₋, e₀, and associated coordinates with respect to this basis, denoted A₊, A₋, A₀, the vector A is:

where the spherical basis vectors can be defined in terms of the Cartesian basis using complex number-valued coefficients in the xy plane:

\mathbf{e}_x -\frac{i}{\sqrt{2}}\mathbf{e}_y \\
     

\mathbf{e}_{-} & = +\frac{1}{\sqrt{2}}\mathbf{e}_x - \frac{i}{\sqrt{2}}\mathbf{e}_y \\

in which $i$ denotes the imaginary unit, and one normal to the plane in the z direction:

$\backslash mathbf\{e\}\_0\; =\; \backslash mathbf\{e\}\_z$

The inverse relations are:

\mathbf{e}_+ + \frac{1}{\sqrt{2}}\mathbf{e_{-}} \\
     

\mathbf{e}_y & = + \frac{i}{\sqrt{2}} \mathbf{e}_+ + \frac{i}{\sqrt{2}}\mathbf{e_{-}} \\ \mathbf{e}_z & = \mathbf{e}_0

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Commutator definition While giving a basis in a 3-dimensional space is a valid definition for a spherical tensor, it only covers the case for when the rank $k$ is 1. For higher ranks, one may use either the commutator, or rotation definition of a spherical tensor. The commutator definition is given below, any operator $T\_q^\{(k)\}$ that satisfies the following relations is a spherical tensor: $$J\_\backslash pm,\; =\; \backslash hbar\; \backslash sqrt\{(k\backslash mp\; q)(k\backslash pm\; q+1)\}T\_\{q\backslash pm\; 1\}^\{(k)\}$$ $$J\_z,\; =\; \backslash hbar\; q\; T\_q^\{(k)\}$$

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Rotation definition Analogously to how the spherical harmonics transform under a rotation, a general spherical tensor transforms as follows, when the states transform under the unitary operator Wigner D-matrix $\backslash mathcal\{D\}(R)$, where is a (3×3 rotation) group element in SO(3). That is, these matrices represent the rotation group elements. With the help of its Lie algebra, one can show these two definitions are equivalent.

$\backslash mathcal\{D\}(R)T\_q^\{(k)\}\backslash mathcal\{D\}^\{\backslash dagger\}(R)\; =\; \backslash sum\_\{q\text{'}\; =\; -k\}^k\; T\_\{q\text{'}\}^\{(k)\}\; \backslash mathcal\{D\}\_\{q\text{'}\; q\}^\{(k)\}$

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Coordinate vectors For the spherical basis, the coordinates are complex-valued numbers A₊, A₀, A₋, and can be found by substitution of () into (), or directly calculated from the inner product , ():

+ \frac{iA_y}{\sqrt{2}} \\
     

A_{-} & = \left\langle \mathbf{A}, \mathbf{e}_- \right\rangle = +\frac{A_x}{\sqrt{2}} + \frac{iA_y}{\sqrt{2}} \\

$A\_0\; =\; \backslash left\backslash langle\; \backslash mathbf\{e\}\_0,\; \backslash mathbf\{A\}\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash mathbf\{e\}\_z,\; \backslash mathbf\{A\}\; \backslash right\backslash rangle\; =\; A\_z$

with inverse relations:

A_+ + \frac{1}{\sqrt{2}} A_{-} \\
     

A_y & = - \frac{i}{\sqrt{2}} A_+ - \frac{i}{\sqrt{2}} A_{-} \\ A_z & = A_0

In general, for two vectors with complex coefficients in the same real-valued orthonormal basis e _i, with the property e _i· e _j = δ_ij, the inner product is:

where · is the usual dot product and the complex conjugate * must be used to keep the magnitude (or "norm") of the vector positive definite.

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Properties (three dimensions)

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Orthonormality The spherical basis is an orthonormal basis, since the inner product , () of every pair vanishes meaning the basis vectors are all mutually orthogonality:

$\backslash left\backslash langle\; \backslash mathbf\{e\}\_+\; ,\; \backslash mathbf\{e\}\_\{-\}\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash mathbf\{e\}\_\{-\}\; ,\; \backslash mathbf\{e\}\_0\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash mathbf\{e\}\_0\; ,\; \backslash mathbf\{e\}\_+\; \backslash right\backslash rangle\; =\; 0$

and each basis vector is a unit vector:

$\backslash left\backslash langle\backslash mathbf\{e\}\_+\; ,\; \backslash mathbf\{e\}\_\{+\}\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\backslash mathbf\{e\}\_\{-\}\; ,\; \backslash mathbf\{e\}\_\{-\}\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\backslash mathbf\{e\}\_0\; ,\; \backslash mathbf\{e\}\_0\; \backslash right\backslash rangle\; =\; 1$

hence the need for the normalizing factors of $1/\backslash !\backslash sqrt\{2\}$.

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Change of basis matrix The defining relations () can be summarized by a transformation matrix U:

$\backslash begin\{pmatrix\}$

\mathbf{e}_+ \\ \mathbf{e}_{-} \\ \mathbf{e}_0 \end{pmatrix} = \mathbf{U}\begin{pmatrix} \mathbf{e}_x \\ \mathbf{e}_y \\ \mathbf{e}_z \end{pmatrix} \,,\quad \mathbf{U} = \begin{pmatrix} - \frac{1}{\sqrt{2}} & - \frac{i}{\sqrt{2}} & 0 \\ + \frac{1}{\sqrt{2}} & - \frac{i}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{pmatrix}\,,

with inverse:

$\backslash begin\{pmatrix\}$

\mathbf{e}_x \\ \mathbf{e}_y \\ \mathbf{e}_z \end{pmatrix} = \mathbf{U}^{-1}\begin{pmatrix} \mathbf{e}_+ \\ \mathbf{e}_{-} \\ \mathbf{e}_0 \end{pmatrix} \,,\quad \mathbf{U}^{-1} = \begin{pmatrix} - \frac{1}{\sqrt{2}} & + \frac{1}{\sqrt{2}} & 0 \\ + \frac{i}{\sqrt{2}} & + \frac{i}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{pmatrix}\,.

It can be seen that U is a unitary matrix, in other words its Hermitian conjugate U^† (complex conjugate and matrix transpose) is also the inverse matrix U⁻¹.

For the coordinates:

$\backslash begin\{pmatrix\}$

A_+ \\ A_{-} \\ A_0 \end{pmatrix} =\mathbf{U}^\mathrm{*} \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} \,,\quad \mathbf{U}^\mathrm{*} = \begin{pmatrix} - \frac{1}{\sqrt{2}} & + \frac{i}{\sqrt{2}} & 0 \\ + \frac{1}{\sqrt{2}} & + \frac{i}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{pmatrix}\,,

and inverse:

$\backslash begin\{pmatrix\}$

A_x \\ A_y \\ A_z \end{pmatrix} = (\mathbf{U}^\mathrm{*})^{-1} \begin{pmatrix} A_+ \\ A_{-} \\ A_0 \end{pmatrix} \,,\quad (\mathbf{U}^\mathrm{*})^{-1} = \begin{pmatrix} - \frac{1}{\sqrt{2}} & + \frac{1}{\sqrt{2}} & 0 \\ - \frac{i}{\sqrt{2}} & - \frac{i}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{pmatrix}\,.

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Cross products Taking of the spherical basis vectors, we find an obvious relation:

$\backslash mathbf\{e\}\_\{q\}\backslash times\backslash mathbf\{e\}\_\{q\}\; =\; \backslash boldsymbol\{0\}$

where q is a placeholder for +, −, 0, and two less obvious relations:

$\backslash mathbf\{e\}\_\{\backslash pm\}\backslash times\backslash mathbf\{e\}\_\{\backslash mp\}\; =\; \backslash pm\; i\; \backslash mathbf\{e\}\_0$

$\backslash mathbf\{e\}\_\{\backslash pm\}\backslash times\backslash mathbf\{e\}\_\{0\}\; =\; \backslash pm\; i\; \backslash mathbf\{e\}\_\{\backslash pm\}$

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Inner product in the spherical basis The inner product between two vectors A and B in the spherical basis follows from the above definition of the inner product:

$\backslash left\backslash langle\; \backslash mathbf\{A\}\; ,\; \backslash mathbf\{B\}\; \backslash right\backslash rangle\; =\; A\_+\; B\_+^\backslash star\; +\; A\_\{-\}B\_\{-\}^\backslash star\; +\; A\_0\; B\_0^\backslash star$

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See also

• Wigner–Eckart theorem
• Wigner D matrix
• 3D rotation group

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General

• S. S. M. Wong (2025). 9783527617913, John Wiley & Sons. . ISBN 9783527617913

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External links

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