"구면조화함수(spherical harmonics)"의 두 판 사이의 차이
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| − | * <math> | + | * <math>l\in \mathbb{Z}_{\geq 0}</math>, <math>-l \leq m \leq l</math>에 대하여, <math>Y_{l}^{m}(\theta,\phi)</math><br> |
| − | * | + | * SO(3) 의 <math>L^2(S^2)</math> 에서의 표현론으로 이해할 수 있다<br> |
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<math>\left( \begin{array}{ccc} 3 & -3 & \frac{1}{8} \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \sin ^3(\theta ) \\ 3 & -2 & \frac{1}{4} \sqrt{\frac{105}{2 \pi }} e^{-2 i \phi } \sin ^2(\theta ) \cos (\theta ) \\ 3 & -1 & \frac{1}{8} \sqrt{\frac{21}{\pi }} e^{-i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right) \\ 3 & 0 & \frac{1}{4} \sqrt{\frac{7}{\pi }} \left(5 \cos ^3(\theta )-3 \cos (\theta )\right) \\ 3 & 1 & -\frac{1}{8} \sqrt{\frac{21}{\pi }} e^{i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right) \\ 3 & 2 & \frac{1}{4} \sqrt{\frac{105}{2 \pi }} e^{2 i \phi } \sin ^2(\theta ) \cos (\theta ) \\ 3 & 3 & -\frac{1}{8} \sqrt{\frac{35}{\pi }} e^{3 i \phi } \sin ^3(\theta ) \end{array} \right)</math> | <math>\left( \begin{array}{ccc} 3 & -3 & \frac{1}{8} \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \sin ^3(\theta ) \\ 3 & -2 & \frac{1}{4} \sqrt{\frac{105}{2 \pi }} e^{-2 i \phi } \sin ^2(\theta ) \cos (\theta ) \\ 3 & -1 & \frac{1}{8} \sqrt{\frac{21}{\pi }} e^{-i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right) \\ 3 & 0 & \frac{1}{4} \sqrt{\frac{7}{\pi }} \left(5 \cos ^3(\theta )-3 \cos (\theta )\right) \\ 3 & 1 & -\frac{1}{8} \sqrt{\frac{21}{\pi }} e^{i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right) \\ 3 & 2 & \frac{1}{4} \sqrt{\frac{105}{2 \pi }} e^{2 i \phi } \sin ^2(\theta ) \cos (\theta ) \\ 3 & 3 & -\frac{1}{8} \sqrt{\frac{35}{\pi }} e^{3 i \phi } \sin ^3(\theta ) \end{array} \right)</math> | ||
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* [[구면(sphere)]], [[라플라시안(Laplacian)]]<br><math>\Delta_{S^2} f = {1 \over \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial f \over \partial \theta} \right) + {1 \over \sin^2 \theta} {\partial^2 f \over \partial \varphi^2}</math><br> | * [[구면(sphere)]], [[라플라시안(Laplacian)]]<br><math>\Delta_{S^2} f = {1 \over \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial f \over \partial \theta} \right) + {1 \over \sin^2 \theta} {\partial^2 f \over \partial \varphi^2}</math><br> | ||
| − | * 라플라시안의<br> | + | * 구면조화함수는 라플라시안의 고유벡터이다<br><math>\Delta_{S^2} Y_{l}^{m}=-l(l+1)Y_{l}^{m}</math><br> |
2011년 12월 3일 (토) 16:19 판
이 항목의 스프링노트 원문주소
개요
- \(l\in \mathbb{Z}_{\geq 0}\), \(-l \leq m \leq l\)에 대하여, \(Y_{l}^{m}(\theta,\phi)\)
- SO(3) 의 \(L^2(S^2)\) 에서의 표현론으로 이해할 수 있다
예
- l=0
\(\left( \begin{array}{ccc} 0 & 0 & \frac{1}{2 \sqrt{\pi }} \end{array} \right)\)
- l=1
\(\left( \begin{array}{ccc} 1 & -1 & \frac{1}{2} \sqrt{\frac{3}{2 \pi }} e^{-i \phi } \sin (\theta ) \\ 1 & 0 & \frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta ) \\ 1 & 1 & -\frac{1}{2} \sqrt{\frac{3}{2 \pi }} e^{i \phi } \sin (\theta ) \end{array} \right)\)
- l=2
\(\left( \begin{array}{ccc} 2 & -2 & \frac{1}{4} \sqrt{\frac{15}{2 \pi }} e^{-2 i \phi } \sin ^2(\theta ) \\ 2 & -1 & \frac{1}{2} \sqrt{\frac{15}{2 \pi }} e^{-i \phi } \sin (\theta ) \cos (\theta ) \\ 2 & 0 & \frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 \cos ^2(\theta )-1\right) \\ 2 & 1 & -\frac{1}{2} \sqrt{\frac{15}{2 \pi }} e^{i \phi } \sin (\theta ) \cos (\theta ) \\ 2 & 2 & \frac{1}{4} \sqrt{\frac{15}{2 \pi }} e^{2 i \phi } \sin ^2(\theta ) \end{array} \right)\)
- l=3
\(\left( \begin{array}{ccc} 3 & -3 & \frac{1}{8} \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \sin ^3(\theta ) \\ 3 & -2 & \frac{1}{4} \sqrt{\frac{105}{2 \pi }} e^{-2 i \phi } \sin ^2(\theta ) \cos (\theta ) \\ 3 & -1 & \frac{1}{8} \sqrt{\frac{21}{\pi }} e^{-i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right) \\ 3 & 0 & \frac{1}{4} \sqrt{\frac{7}{\pi }} \left(5 \cos ^3(\theta )-3 \cos (\theta )\right) \\ 3 & 1 & -\frac{1}{8} \sqrt{\frac{21}{\pi }} e^{i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right) \\ 3 & 2 & \frac{1}{4} \sqrt{\frac{105}{2 \pi }} e^{2 i \phi } \sin ^2(\theta ) \cos (\theta ) \\ 3 & 3 & -\frac{1}{8} \sqrt{\frac{35}{\pi }} e^{3 i \phi } \sin ^3(\theta ) \end{array} \right)\)
내적
\(\int _0^{2\pi }\int _0^{\pi }Y_l^m(\theta ,\phi ){}^*Y_L^M(\theta ,\phi ) \sin (\theta )d\theta d\phi =\delta _{l,L}\delta _{m,M}.\)
단위구면의 라플라시안
- 구면(sphere), 라플라시안(Laplacian)
\(\Delta_{S^2} f = {1 \over \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial f \over \partial \theta} \right) + {1 \over \sin^2 \theta} {\partial^2 f \over \partial \varphi^2}\) - 구면조화함수는 라플라시안의 고유벡터이다
\(\Delta_{S^2} Y_{l}^{m}=-l(l+1)Y_{l}^{m}\)
역사
메모
관련된 항목들
수학용어번역
사전 형태의 자료
- http://ko.wikipedia.org/wiki/구면조화함수
- http://en.wikipedia.org/wiki/spherical_harmonics
- http://en.wikipedia.org/wiki/Associated_Legendre_polynomials
- http://en.wikipedia.org/wiki/Table_of_spherical_harmonics
- http://www.wolframalpha.com/input/?i=spherical+harmonics
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
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