# 구면조화함수(spherical harmonics)

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## 개요

• 3차원 공간에서 정의된 조화다항식의 구면에 제한(restrict)하여 얻어지는 구면 위에 정의되는 함수를 일반적으로 구면조화함수라 함
• 3차원 회전군 SO(3)의 $$L^2(S^2)$$ 에서의 표현론으로 이해
• $$S^2=SO(3)/SO(2)$$
• 양자역학에서 원자모형을 이해하는데 중요한 역할

## 정의

• $$l\in \mathbb{Z}_{\geq 0}$$, $$-l \leq m \leq l$$에 대하여, $$Y_{l}^{m}(\theta,\phi)$$을 다음과 같이 정의

$Y_l^m(\theta ,\phi )=\sqrt{(2l+1)/(4\pi )}\sqrt{(l-m)!/(l+m)!}P_l^m(\cos (\theta ))e^{im\phi }$ 여기서 $$P_l^m(x)$$는 르장드르 다항식(associated Legendre polynomials)

## 테이블

• 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)$\Delta_{S^2} f = {\partial^2 f \over \partial \theta^2} +\cot\theta {\partial f \over \partial \theta} + \frac{1}{ \sin^2 \theta}{\partial^2 f \over \partial \phi^2}$
• 구면조화함수는 라플라시안의 고유벡터이며, 고유치는 $$-l(l+1)$$ 이다$\Delta_{S^2} Y_{l}^{m}=-l(l+1)Y_{l}^{m}$

## 각운동량 연산자

여기서

$$L^2=-\hbar ^2 \left(\frac{1}{\sin ^2(\theta )}\frac{\partial^2}{\partial \phi^2}+\frac{1}{\sin (\theta )} \frac{\partial }{\partial \theta }\left(\sin (\theta ) \frac{\partial}{\partial \theta }\right)\right)$$

$$L_{z}=-i \hbar \frac{\partial}{\partial \phi }$$

## 예

• $$l=3,m=1$$ 인 경우$Y_{3}^{1}(\theta,\phi)=-\frac{1}{8} \sqrt{\frac{21}{\pi }} e^{i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)$
• $$L^2 Y_{3}^{1}(\theta,\phi)=12\hbar^2Y_{3}^{1}$$
• $$L_{z}Y_{3}^{1}(\theta,\phi)=\hbar Y_{3}^{1}$$

## 3-j 기호(3-j symbols)의 관계

\begin{align} & {} \quad \int Y_{l_1}^{m_1}(\theta,\varphi)Y_{l_2}^{m_2}(\theta,\varphi)Y_{l_3}^{m_3}(\theta,\varphi)\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi \\ & = \sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \begin{pmatrix} l_1 & l_2 & l_3 \\[8pt] 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \end{align}

## 관련논문

• Cammarota, Valentina, and Igor Wigman. “Fluctuations of the Total Number of Critical Points of Random Spherical Harmonics.” arXiv:1510.00339 [math], October 1, 2015. http://arxiv.org/abs/1510.00339.
• Achour, J. Ben, E. Huguet, J. Queva, and J. Renaud. ‘Explicit Vector Spherical Harmonics on the 3-Sphere’. arXiv:1505.03426 [gr-Qc, Physics:hep-Th, Physics:math-Ph], 13 May 2015. http://arxiv.org/abs/1505.03426.
• Nazarov, Fedor, and Mikhail Sodin. “On the Number of Nodal Domains of Random Spherical Harmonics.” arXiv:0706.2409 [math-Ph], June 18, 2007. http://arxiv.org/abs/0706.2409.

## 메타데이터

### Spacy 패턴 목록

• [{'LOWER': 'spherical'}, {'LEMMA': 'harmonic'}]

## 노트

### 말뭉치

1. Spherical harmonic functions arise when the spherical coordinate system is used.
2. As the general function shows above, for the spherical harmonic where $$l = m = 0$$, the bracketed term turns into a simple constant.
3. It appears that for every even, angular QM number, the spherical harmonic is even.
4. Consider the question of wanting to know the expectation value of our colatitudinal coordinate $$\theta$$ for any given spherical harmonic with even-$$l$$.
5. (1 - x^{2})^{\tiny\dfrac{|m|}{2}}\dfrac{d^{|m|}}{dx^{|m|}}P_{l}(x)\) Using these recurrence relations, write the spherical harmonic $$Y_{1}^{1}(\theta,\phi)$$.
6. Thus, the object surface can be described via expanding these three spherical functions using spherical harmonics.
7. B Spherical Harmonics SPHERICAL harmonics are a frequency-space basis for representing functions dened over the sphere.
8. Spherical harmonics arise in many physical problems ranging from the computation of atomic electron congurations to the representation of gravitational and magnetic elds of planetary bodies.
9. In this appendix, we briey review the spherical harmonics as they relate to computer graphics.
10. As their name suggests, the spherical harmonics are an innite set of harmonic functions dened on the sphere.
11. The spherical harmonics are dened as the wave functions of angular mo- mentum eigenstates Y m l (, ) = h, |l, mi.
12. The latter equation is easy to solve: the azimuth dependence of the spherical harmonics must be eim.
13. We can take the same strategy for the spherical harmonics.
14. You can verify the orthonormality of spherical harmonics explicitly.
15. One concludes that the spherical harmonics in the solution for the electron wavefunction in the hydrogen atom identify the angular momentum of the electron.
16. As stated, spherical harmonics routinely arise in physical settings due to the prevalence of the Laplacian in many physical equations.
17. Find the potential in terms of spherical harmonics in all of space ( r < R (r<R (r<R and r > R ) .
18. The angular dependence at r = R r=R r=R solved for above in terms of spherical harmonics is therefore the angular dependence everywhere.
19. The completeness property of the spherical harmonics implies that any well-behaved function of and can be written as f (, ) = amY m (, ) .
20. r 3 4 r 1 4r 15 2 15 8 5 4 r 1 2r 3 The corresponding spherical harmonics for negative values of m are obtained using eq.
21. , (13) which relates the Legendre polynomials to the spherical harmonics with m = 0.
22. (14) and (15) imply that: That is, the spherical harmonics are eigenfunctions of the dierential operator ~L2, with corresponding eigenvalues ( + 1), for = 0, 1, 2, 3, . .
23. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right.
24. Spherical harmonics originates from solving Laplace's equation in the spherical domains.
25. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions.
26. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions.
27. The spherical harmonics are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present.
28. Spherical harmonics satisfy the spherical harmonic differential equation, which is given by the angular part of Laplace's equation in spherical coordinates.
29. Sometimes (e.g., Arfken 1985), the Condon-Shortley phase is prepended to the definition of the spherical harmonics.
30. Nevertheless, given the importance of spherical harmonics in many applications, Spherefun allows one to compute with spherical harmonics.
31. In this and the next four sections we discuss some properties of spherical harmonics and show how Spherefun can be used to easily verify them.
32. The command sphharm constructs a spherical harmonic of a given degree and order.
33. Black contour lines have been included indicating the zero curves of each spherical harmonic, which highlights their transition from positive to negative values.
34. Moreover, being real, they have half the memory requirement of complex spherical harmonics.
35. The basic properties of RSH can be easily derived from the properties of complex spherical harmonics by means of \ref{eq:Definition_real_harmonics}.
36. In a similar way, the electrostatic potentials can be expanded in a real spherical harmonics basis set.
37. We use a specific set of spherical harmonics, denoted Y^m_l(\theta,\phi) called Laplace's spherical harmonics.
38. Geophysical analyses are often performed in spherical geometry and require the use of spherical harmonic functions to express observables or physical quantities.
39. When expanded to high degree, the accuracy and speed of the spherical harmonic transforms and reconstructions are of paramount importance.
40. SHTools is a time and user‐tested open‐source archive of both Fortran 95 and Python routines for performing spherical harmonic analyses.
41. The spherical‐harmonic transforms are proven to be fast and accurate for spherical harmonic degrees up to 2800.
42. This relation is easily extended to spherical geometry using the orthogonality properties of the spherical harmonic functions.
43. $$S$$ is the total power of the function at spherical harmonic degree $$l$$, which in pyshtools is called the power per degree $$l$$.
44. Alternatively, one can calculate the average power per coefficient at spherical harmonic degree $$l$$, which in pyshtools is referred to as the power per $$lm$$.
45. \end{equation} One can also calculate the power from all angular orders over an infinitesimal logarithmic spherical harmonic degree band $$d \log_a l$$, where $$a$$ is the logarithmic base.
46. For each spherical harmonic, we plot its value on the surface of a sphere, and then in polar.
47. This is especially true when it comes to rotations of spherical harmonics (much of the literature is math-dense and contains errata).
48. This library is a collection of useful functions for working with spherical harmonics.
49. It can be computed quickly by estimating the standard diffuse cosine-lobe as a vector of coefficients, and the environment as spherical harmonics.
50. Rotation - Object type that computes the transformation matrices that suitably transform spherical harmonic coefficients given a quaternion rotation.
51. In quantum mechanics spherical harmonics appear as eigenfunctions of (squared) orbital angular momentum.
52. The geodesy and magnetics communities never include the Condon-Shortley phase factor in their definitions of the spherical harmonic functions.
53. For example, a space with an odd number of dimensions (2ℓ+1) can be constructed from the spherical harmonics Y ℓ m, and their transformations under infinitesimal rotations.
54. 6 Direction of arrival estimation in the spherical harmonic domain using subspace pseudointensity vectors ,” IEEE/ACM Trans.
55. To improve the accuracy of DOA estimation, an augmented intensity vector is proposed by exploiting higher order spherical harmonics.
56. In such instances, spherical harmonics can play a role because they cover the entire space together with the radial functions.
57. Therefore, if we can generate a spherical harmonic decomposition of the spatial sound intensity, the sound intensity at any point in space is readily available.
58. Spherical harmonics are functions that arise in physics and mathematics in the study of the same kind of systems as for which spherical polar coordinates (r, θ, and φ) are useful.
59. The spherical harmonic functions depend on the spherical polar angles θ and φ and form an (infinite) complete set of orthogonal, normalizable functions.
60. As stated, the spherical harmonics—almost always written as Y m ℓ (θ, φ)—form an orthogonal and complete set.
61. The spherical harmonic functions can be used to describe a function of θ and φ in the form of a linear expansion.
62. Our spherical harmonics therefore span L2.
63. We would like to nd the conditions on P m n to make (13) a set of (smooth) spherical harmonics.
64. The aliasing of spherical harmonics follows the Fourier rules in , and is similar to the reective aliasing in .
65. It provides a brief overview of spherical harmonics (SH) and discusses several ways they can be used in interactive graphics and problems that might arise.
66. While spherical harmonics may seem somewhat daunting, they are actually straightforward.
67. Check the condition of the transformation between DWIs and spherical harmonics.
68. The data are transformed not to the spherical harmonic basis, but directly to the zonal spherical harmonic basis (this is the spherical harmonic basis containing only the m = 0 terms).
69. The spherical harmonic series therefore provides a compact represention for smooth functions on the sphere.
70. Spherical harmonics are special functions defined on the surface of a sphere.
71. That question is " I just want a function for the sphere (in spherical coordinates so I can expand it in terms of the spherical harmonics).
72. But that to say in difficult problems the use of spherical harmonics is laborious is not to slight the method, because any other accurate treatment would be still more difficult.
73. At the very least, spherical harmonic analysis provides a way to synthesize from a scatter of discrete measurements on a sphere an equation applicable to the entire sphere.
74. In particular, the various terms of a spherical harmonic expansion are sometimes related (with caution) to specific physical phenomena.
75. In places they can be rotated to the true spherical harmonics .
76. (10) gives the transformation of an eigenvector between real and spherical harmonics.
77. Each term can be re-expressed in spherical harmonics through the rotation in the first expression, Eq.
78. We consider a pair linear combinations of spherical harmonics derived from the operator acting on a single function.
79. The spherical harmonic is evaluated at the spherical coordinate (theta,phi) on the unit sphere in S^2.
80. Computes the divergent (irrotational) wind components for a fixed grid via spherical harmonics.
81. dv2uvf Computes the divergent (irrotational) wind components for a fixed grid via spherical harmonics.
82. dv2uvF_Wrap Computes the divergent (irrotational) wind components for a fixed grid via spherical harmonics and retains metadata.
83. dv2uvg Computes the divergent (irrotational) wind components for a gaussian grid via spherical harmonics.
84. Spherical harmonics basis functions are used for parametrisation.
85. This involves least squares fitting of spherical harmonics basis functions to the surface mesh.
86. This paper proposes a technique for parametrising 3D meshes for remeshing which uses the theory of spherical harmonics to approximate a continuous surface.
87. Spherical harmonics are a natural basis for representing functions defined over spherical and hemispherical domains.
88. Several normalizations for the spherical harmonics exist (details on wikipedia).
89. Use shtns_create with SHT_REAL_NORM to use a "real" spherical harmonic normalization.
90. This is the usual "real" spherical harmonics, if one takes the complex conjugate of the coefficients.

## 메타데이터

### Spacy 패턴 목록

• [{'LOWER': 'spherical'}, {'LEMMA': 'harmonic'}]