Rotation of a vector is easy and straight forward, just apply a rotation matrix on the vector. But rotating a function may be tricky, we need to transform the coordinate one by one.

When rotating spherical harmonic, the thing becomes get. We can treat the spherical harmonic as a eigen state of angular momentum operator. The problem will becomes easy.

Recall that, the rotation operator for a eigen-state is

,

The matrix element is the Wigner D-matrix,

The spherical harmonic is

Thus, the rotation of spherical harmonic is

or

The above picture is rotated by .

We can see, the spherical harmonic formed a close set under Wigner D-matrix, that combination of them are themselves.

The derivation is follow. Notice that

Thus,

We can use the identity

Than

or

It is easy to show that

or

Note that

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