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I am looking for some direction to books or papers which may help me,

When you solve the Helmholtz equation you end up with vector spherical harmonics as solutions. The Helmholtz equation is invarient under rotations which means that rotations of these solutions are also solutions.

I am looking for a way to rotate these solutions, for example around the y axis by some angle β.

The solutions look like:

So far all that I have been able to find is that rotations of these solutions can be expressed as a linear combination of them over different l's. What I am unsure about is what the weighting is of this linear combination. I have found some people state that it is the Wigner D function (with an unhelpful reference), but after doing the calculations it doesn't seem like these work as I would have expected.

It's hard for me to distinguish between bad coding and bad intuition at the moment because I haven't found much information about such rotations online.

Is the Wigner D function the correct waiting? Are there some other rotation matrix elements that I should be using as the weighting? Does anyone know of anywhere I could find some more information to learn this properly?

Any help is greatly appreciated,

Thanks

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# Rotating harmonics

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