Rotation of spherical harmonics

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Homework Statement
Rotate the spherical harmonic $$\ket{l=2, m=1}=Y_{2, 1}$$ an angle of π/4 about the y-axis.
Relevant Equations
$$\sum_{m'=-l}^{l} {d^{(l)}}_{m, m'} Y_{l, m'}$$
I tried using the Wigner matrices:

$$\sum_{m'=-2}^{2} {d^{(2)}}_{1m'} Y_{2; m'}={d^{(2)}}_{1 -2} Y_{2; -2} + {d^{(2)}}_{1 -1} Y_{2; -1} + ...= -\frac{1-\cos(\beta)}{2} \sin(\beta) \sqrt{\frac{15}{32 \pi}} \sin^2(\theta) e^{-i \phi} + ...$$

where $$\beta=\frac{\pi}{4}$$. But I don't know if this is ok since $$\beta$$ is an Euler angle while $$\theta$$ and $$\phi$$ are not. If this is not right, what should I do?
 
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  • #2
The angle ##\beta## will go away as it is replaced by the value of the rotation, leaving a function of ##(\theta,\phi)##, which is what you want.
 
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1. What is the definition of rotation of spherical harmonics?

The rotation of spherical harmonics is a mathematical operation that describes how a spherical harmonic function changes when the coordinate system is rotated. It is used to represent the orientation of an object in three-dimensional space.

2. How is the rotation of spherical harmonics calculated?

The rotation of spherical harmonics is calculated using a rotation matrix, which is a mathematical tool that describes the transformation of coordinates from one coordinate system to another. This matrix is applied to the spherical harmonic function to determine its new orientation.

3. What are the applications of rotation of spherical harmonics?

The rotation of spherical harmonics has various applications in fields such as physics, computer graphics, and signal processing. It is used to describe the orientation of molecules, the motion of planets, and the rotation of objects in 3D animations.

4. Can the rotation of spherical harmonics be reversed?

Yes, the rotation of spherical harmonics can be reversed by applying the inverse of the rotation matrix. This will return the spherical harmonic function to its original orientation.

5. Are there any limitations to the rotation of spherical harmonics?

One limitation of the rotation of spherical harmonics is that it only applies to objects with spherical symmetry. It cannot be used to describe the rotation of irregularly shaped objects. Additionally, the rotation of spherical harmonics is limited to three-dimensional space and cannot be extended to higher dimensions.

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