A Confused about multipole expansion of vector potential

AI Thread Summary
The discussion revolves around understanding the vector spherical harmonics (VSH) and their application in multipole expansions of vector potentials. The original poster is confused about the definitions of VSH in different sources and how to derive the hyperfine interaction operator from the provided equations. Key points include the need for three types of VSH for multipole decomposition and their orthogonality properties. There is also frustration expressed regarding the lack of clarity in the referenced paper and the expectation to consult additional resources. The poster seeks further clarification on the use of specific terms and notation in the equations.
kelly0303
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Hello! I found an expression in this paper (eq. 1) for the multipole expansion of the vector potential. I am not sure I understand what form do the vector spherical harmonics (VSH) have. Also, for example, the usual hyperfine interaction operator is given by ##\frac{\mathbf{\mu}\cdot(\mathbf{r}\times \mathbf{\alpha})}{r^3}##. I am not sure how to get back to this expression using equation 1 (or 2), for k=1. On Wikipedia it seems like VSH are defined as ##Y_{lm}\hat{r}##, while in the reference they mention in the paper it would be ##\frac{1}{\sqrt{J(J+1)}}\mathbf{L}Y_{JM}##, where ##\mathbf{L}## is the orbital angular momentum operator. I tried using both and still didn't get back the original formula. Can someone help me with this? Thank you!
 
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For a multipole decomposition of a general vector field you need three kinds of vector-spherical harmonics. Written in standard spherical coordinates they are
$$\vec{\Psi}_{lm}(\vartheta,\varphi)=r \vec{\nabla} \mathrm{Y}_{lm}(\vartheta,\varphi),$$
$$\vec{\Phi}_{lm}(\vartheta,\varphi)=\vec{r} \times \vec{\nabla} \mathrm{Y}_{lm}(\vartheta,\varphi),$$
and
$$\vec{\mathrm{Y}}_{lm}(\vartheta,\varphi)=\vec{e}_r \text{Y}_{lm}(\vartheta,\varphi).$$
These are all mutually orthogonal to each other under the scalar product on the unit sphere
$$\langle \vec{V}_1|\vec{V}_2 \rangle=\int_{0}^{\pi} \mathrm{d} \vartheta \int_0^{2 \pi} \mathrm{d} \varphi \sin \vartheta \vec{V}_1^*(\vartheta,\varphi) \cdot \vec{V}_2(\vartheta,\varphi)$$
and normlized according to
$$\langle \vec{\Psi}_{lm}|\vec{\Psi}_{l'm'} \rangle=l(l+1) \delta_{ll'} \delta_{mm'},$$
$$\langle \vec{\Phi}_{lm}|\vec{\Phi}_{l'm'} \rangle=l(l+1) \delta_{ll'} \delta_{mm'},$$
$$\langle \vec{\mathrm{Y}}_{lm}|\vec{\mathrm{Y}}_{l'm'} \rangle=\delta_{ll'} \delta_{mm'}.$$
 
vanhees71 said:
For a multipole decomposition of a general vector field you need three kinds of vector-spherical harmonics. Written in standard spherical coordinates they are
$$\vec{\Psi}_{lm}(\vartheta,\varphi)=r \vec{\nabla} \mathrm{Y}_{lm}(\vartheta,\varphi),$$
$$\vec{\Phi}_{lm}(\vartheta,\varphi)=\vec{r} \times \vec{\nabla} \mathrm{Y}_{lm}(\vartheta,\varphi),$$
and
$$\vec{\mathrm{Y}}_{lm}(\vartheta,\varphi)=\vec{e}_r \text{Y}_{lm}(\vartheta,\varphi).$$
These are all mutually orthogonal to each other under the scalar product on the unit sphere
$$\langle \vec{V}_1|\vec{V}_2 \rangle=\int_{0}^{\pi} \mathrm{d} \vartheta \int_0^{2 \pi} \mathrm{d} \varphi \sin \vartheta \vec{V}_1^*(\vartheta,\varphi) \cdot \vec{V}_2(\vartheta,\varphi)$$
and normlized according to
$$\langle \vec{\Psi}_{lm}|\vec{\Psi}_{l'm'} \rangle=l(l+1) \delta_{ll'} \delta_{mm'},$$
$$\langle \vec{\Phi}_{lm}|\vec{\Phi}_{l'm'} \rangle=l(l+1) \delta_{ll'} \delta_{mm'},$$
$$\langle \vec{\mathrm{Y}}_{lm}|\vec{\mathrm{Y}}_{l'm'} \rangle=\delta_{ll'} \delta_{mm'}.$$
Thank you for the reply. But I am not sure I understand how to use this for the given expression. Is ##C_{k,\mu}^{(0)}(\hat{r})## a linear combination of the 3 terms you mentioned above? Also, what is the ##(0)## standing for?
 
I hate non-selfcontained papers :-(. Obviously they expect that you have the cited book at hand and look it up. Just laziness! I don't have the book at hand unfortunately.
 
vanhees71 said:
I hate non-selfcontained papers :-(. Obviously they expect that you have the cited book at hand and look it up. Just laziness! I don't have the book at hand unfortunately.
Ah I see it's not even a universal definition... I found the book here (please let me know if you can't access it). The section is 1.5.2, I would appreciate any insight from you as I am still confused after reading it.
 
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