Timeline for How to prove that spherical and Cartesian $l$th multipole moments have the same number of independent components?
Current License: CC BY-SA 4.0
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| May 22, 2023 at 9:24 | history | edited | user3384598 | CC BY-SA 4.0 |
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| May 20, 2023 at 10:06 | comment | added | user3384598 | Exactly, that's the idea. However, this reasoning suggests that you need only the solutions of the same order $\alpha_x+\alpha_y+\alpha_z=l$ -- something fishy is going on, but I'm not sure where the issue is! Finally, the spherical components of a tensor (indexed like spherical harmonics) are defined as a representation that "behaves like" spherical harmonics when undergoing rotations. (See, for example, §3.11 in J. J. Sakurai, Modern Quantum Mechanics, 2nd ed.) | |
| May 19, 2023 at 18:40 | comment | added | Ulysses Zhan | Thank you for answering! Can I understand your argument as the following? The field $f_{(\alpha_x,\alpha_y,\alpha_z)}$ due to each multipole moment component is expressable using the linear combination of $2l+1$ linearly-independent functions $f_{(l,m)}$, so the multipole momenet components only have $2l+1$ independent ones. Also, could you please explain about the "spherical tensors" that Jackson mentioned? | |
| May 19, 2023 at 12:53 | review | Late answers | |||
| May 19, 2023 at 13:40 | |||||
| S May 19, 2023 at 12:32 | review | First answers | |||
| May 19, 2023 at 13:52 | |||||
| S May 19, 2023 at 12:32 | history | answered | user3384598 | CC BY-SA 4.0 |