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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 rather than .
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:
(2)
where the spherical basis vectors can be defined in terms of the Cartesian basis using complex-valued coefficients in the xy plane:[1]
(3A)
in which denotes the imaginary unit, and one normal to the plane in the z direction:
The inverse relations are:
(3B)
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 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 that satisfies the following relations is a spherical tensor:
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 unitaryWigner D-matrix, where R 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.
For the spherical basis, the coordinates are complex-valued numbers A+, A0, A−, and can be found by substitution of (3B) into (1), or directly calculated from the inner product ⟨, ⟩ (5):
(4A)
with inverse relations:
(4B)
In general, for two vectors with complex coefficients in the same real-valued orthonormal basis ei, with the property ei·ej = δij, the inner product is: