Сферни хармоници у математици представљају угаони део решења Лапласове једначине у сферним координатама.
Сферне хармонике је први 1782 . увео Пјер Симон Лаплас , а облика су:
Y
ℓ
m
(
θ
,
φ
)
=
(
2
ℓ
+
1
)
4
π
(
ℓ
−
m
)
!
(
ℓ
+
m
)
!
P
ℓ
m
(
cos
θ
)
e
i
m
φ
{\displaystyle Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }}
и решење су једначине:
1
sin
θ
d
d
θ
(
sin
θ
d
Y
ℓ
m
d
θ
)
+
(
l
(
l
+
1
)
−
m
2
sin
2
θ
)
Y
ℓ
m
=
0
{\displaystyle {\frac {1}{\sin {\theta }}}{\frac {d}{d\theta }}\left(\sin {\theta }{\frac {dY_{\ell }^{m}}{d\theta }}\right)+(l(l+1)-{\frac {m^{2}}{\sin ^{2}{\theta }}})Y_{\ell }^{m}=0}
уреди
Лапласова једначина у сферним координатама има облик:
∇
2
f
=
1
r
2
∂
∂
r
(
r
2
∂
f
∂
r
)
+
1
r
2
sin
θ
∂
∂
θ
(
sin
θ
∂
f
∂
θ
)
+
1
r
2
sin
2
θ
∂
2
f
∂
φ
2
=
0.
{\displaystyle \nabla ^{2}f={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial f \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial f \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}=0.}
Једначину решавамо сепарацијом варијабли претпостављајући решење облика:
f
(
r
,
ϑ
,
φ
)
=
R
(
r
)
Y
(
ϑ
,
φ
)
{\displaystyle f(r,\vartheta ,\varphi )=R(r)Y(\vartheta ,\varphi )}
Сепарацијом варијабли добија се:
Δ
R
(
r
)
Y
(
ϑ
,
φ
)
=
Y
(
ϑ
,
φ
)
Δ
r
R
(
r
)
+
R
(
r
)
r
2
Δ
ϑ
,
φ
Y
(
ϑ
,
φ
)
=
0
{\displaystyle \Delta R(r)Y(\vartheta ,\varphi )=Y(\vartheta ,\varphi )\Delta _{r}R(r)+{\frac {R(r)}{r^{2}}}\Delta _{\vartheta ,\varphi }Y(\vartheta ,\varphi )=0}
Множећи са
r
2
{\displaystyle r^{2}}
R
(
r
)
Y
(
ϑ
,
φ
)
{\displaystyle R(r)Y(\vartheta ,\varphi )}
r
2
Δ
r
R
(
r
)
R
(
r
)
+
Δ
ϑ
,
φ
Y
(
ϑ
,
φ
)
Y
(
ϑ
,
φ
)
=
0
{\displaystyle {\frac {r^{2}\Delta _{r}R(r)}{R(r)}}+{\frac {\Delta _{\vartheta ,\varphi }Y(\vartheta ,\varphi )}{Y(\vartheta ,\varphi )}}=0}
односно добијају се две једначине:
1
R
d
d
r
(
r
2
d
R
d
r
)
=
λ
,
1
Y
1
sin
θ
∂
∂
θ
(
sin
θ
∂
Y
∂
θ
)
+
1
Y
1
sin
2
θ
∂
2
Y
∂
φ
2
=
−
λ
.
{\displaystyle {\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)=\lambda ,\qquad {\frac {1}{Y}}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial Y}{\partial \theta }}\right)+{\frac {1}{Y}}{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}Y}{\partial \varphi ^{2}}}=-\lambda .}
Угаона једначина
(
∂
2
∂
ϑ
2
+
cos
ϑ
sin
ϑ
∂
∂
ϑ
+
1
sin
2
ϑ
∂
2
∂
φ
2
)
Y
(
ϑ
,
φ
)
=
−
λ
Y
(
ϑ
,
φ
)
{\displaystyle \left({\frac {\partial ^{2}}{\partial \vartheta ^{2}}}+{\frac {\cos \vartheta }{\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}+{\frac {1}{\sin ^{2}\vartheta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}\right)Y(\vartheta ,\varphi )=-\lambda Y(\vartheta ,\varphi )}
може даље да се сепарира по две варијабле:
Y
(
ϑ
,
φ
)
=
Θ
(
ϑ
)
Φ
(
φ
)
{\displaystyle Y(\vartheta ,\varphi )=\Theta (\vartheta )\Phi (\varphi )}
Одатле се добија:
sin
2
ϑ
Θ
(
ϑ
)
(
∂
2
∂
ϑ
2
+
cos
ϑ
sin
ϑ
∂
∂
ϑ
)
Θ
(
ϑ
)
+
sin
2
(
ϑ
)
λ
)
⏟
m
2
=
−
1
Φ
(
φ
)
∂
2
∂
φ
2
Φ
(
φ
)
⏟
m
2
{\displaystyle \underbrace {{\frac {\sin ^{2}\vartheta }{\Theta (\vartheta )}}\left({\frac {\partial ^{2}}{\partial \vartheta ^{2}}}+{\frac {\cos \vartheta }{\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}\right)\Theta (\vartheta )+\sin ^{2}(\vartheta )\lambda )} _{m^{2}}=\underbrace {-{\frac {1}{\Phi (\varphi )}}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}\Phi (\varphi )} _{m^{2}}}
тј. две једначине:
1
Φ
(
φ
)
d
2
Φ
(
φ
)
d
φ
2
=
−
m
2
{\displaystyle {\frac {1}{\Phi (\varphi )}}{\frac {d^{2}\Phi (\varphi )}{d\varphi ^{2}}}=-m^{2}}
λ
sin
2
(
θ
)
+
sin
(
θ
)
Θ
(
θ
)
d
d
θ
[
sin
(
θ
)
d
Θ
d
θ
]
=
m
2
{\displaystyle \lambda \sin ^{2}(\theta )+{\frac {\sin(\theta )}{\Theta (\theta )}}{\frac {d}{d\theta }}\left[\sin(\theta ){\frac {d\Theta }{d\theta }}\right]=m^{2}}
Решење прве једначине је:
Φ
m
(
φ
)
=
A
exp
(
i
m
φ
)
{\displaystyle \Phi _{m}(\varphi )=A\exp(im\varphi )}
Да би друга једначина имала решење мора бити задовољено
λ
=
l
(
l
+
1
)
{\displaystyle \lambda =l(l+1)}
Коначно за угао
θ
{\displaystyle \theta }
1
sin
θ
d
d
θ
(
sin
θ
d
Θ
l
m
d
θ
)
−
m
2
sin
2
θ
Θ
l
m
+
l
(
l
+
1
)
Θ
l
m
=
0
{\displaystyle {\frac {1}{\sin {\theta }}}{\frac {d}{d\theta }}\left(\sin {\theta }{\frac {d\Theta _{lm}}{d\theta }}\right)-{\frac {m^{2}}{\sin ^{2}{\theta }}}\Theta _{lm}+l(l+1)\Theta _{lm}=0}
Уведемо ли супституцију
x
=
cos
(
θ
)
{\displaystyle x=\cos(\theta )}
(
1
−
x
2
)
Θ
″
−
2
x
Θ
′
+
(
ℓ
[
ℓ
+
1
]
−
m
2
1
−
x
2
)
Θ
=
0
,
{\displaystyle (1-x^{2})\,\Theta ''-2x\Theta '+\left(\ell [\ell +1]-{\frac {m^{2}}{1-x^{2}}}\right)\,\Theta =0,\,}
односно једначина чије решење су придружени Лежандрови полиноми
P
l
m
(
cos
ϑ
)
{\displaystyle P_{lm}(\cos \vartheta )}
∫
0
π
|
Θ
l
m
(
ϑ
)
|
2
sin
(
ϑ
)
d
ϑ
=
1
{\displaystyle \int _{0}^{\pi }|\Theta _{lm}(\vartheta )|^{2}\sin(\vartheta )\mathrm {d} \vartheta =1}
Θ
l
m
(
ϑ
)
=
2
l
+
1
2
⋅
(
l
−
m
)
!
(
l
+
m
)
!
P
l
m
(
cos
ϑ
)
{\displaystyle \Theta _{lm}(\vartheta )={\sqrt {{\frac {2l+1}{2}}\cdot {\frac {(l-m)!}{(l+m)!}}}}\,\,P_{lm}(\cos \vartheta )}
Исто тако треба да се нормира и по другом углу
∫
0
2
π
|
Φ
m
(
φ
)
|
2
d
φ
=
1
{\displaystyle \int _{0}^{2\pi }|\Phi _{m}(\varphi )|^{2}\mathrm {d} \varphi =1}
Φ
m
(
φ
)
=
1
2
π
exp
(
i
m
φ
)
,
m
∈
Z
{\displaystyle \Phi _{m}(\varphi )={\frac {1}{\sqrt {2\pi }}}\exp(im\varphi ),\quad m\in \mathbb {Z} }
Заједничко угаоно решење је онда управо функција, коју називамо сферни хармоник:
Y
ℓ
m
(
θ
,
φ
)
=
(
2
ℓ
+
1
)
4
π
(
ℓ
−
m
)
!
(
ℓ
+
m
)
!
P
ℓ
m
(
cos
θ
)
e
i
m
φ
{\displaystyle Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }}
Сферни хармоници су ортогонални:
∫
θ
=
0
π
∫
φ
=
0
2
π
Y
ℓ
m
Y
ℓ
′
m
′
∗
d
Ω
=
δ
ℓ
ℓ
′
δ
m
m
′
,
{\displaystyle \int _{\theta =0}^{\pi }\int _{\varphi =0}^{2\pi }Y_{\ell }^{m}\,Y_{\ell '}^{m'*}\,d\Omega =\delta _{\ell \ell '}\,\delta _{mm'},}
Задовољавају релацију потпуности:
∑
l
=
0
∞
∑
m
=
−
l
l
Y
l
m
∗
(
ϑ
′
,
φ
′
)
Y
l
m
(
ϑ
,
φ
)
=
δ
(
φ
−
φ
′
)
δ
(
cos
ϑ
−
cos
ϑ
′
)
{\displaystyle \sum _{l=0}^{\infty }\sum _{m=-l}^{l}Y_{lm}^{*}(\vartheta ',\varphi ')\,Y_{lm}(\vartheta ,\varphi )=\delta (\varphi -\varphi ')\delta (\cos {\vartheta }-\cos {\vartheta '})}
Осим тога у случају трансформација вреди:
Y
l
m
(
π
−
ϑ
,
π
+
φ
)
=
(
−
1
)
l
⋅
Y
l
m
(
ϑ
,
φ
)
{\displaystyle Y_{lm}(\pi -\vartheta ,\pi +\varphi )=(-1)^{l}\cdot Y_{lm}(\vartheta ,\varphi )}
Y
l
,
−
m
(
ϑ
,
φ
)
=
(
−
1
)
m
⋅
Y
l
m
∗
(
ϑ
,
φ
)
{\displaystyle Y_{l,-m}(\vartheta ,\varphi )=(-1)^{m}\cdot Y_{lm}^{*}(\vartheta ,\varphi )}
Интеграл три сферна хармоника дат је преко 3-jm симбола :
∫
Y
l
1
m
1
(
θ
,
φ
)
Y
l
2
m
2
(
θ
,
φ
)
Y
l
3
m
3
(
θ
,
φ
)
sin
θ
d
θ
d
φ
=
(
2
l
1
+
1
)
(
2
l
2
+
1
)
(
2
l
3
+
1
)
4
π
(
l
1
l
2
l
3
0
0
0
)
(
l
1
l
2
l
3
m
1
m
2
m
3
)
{\displaystyle {\begin{aligned}&{}\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{aligned}}}
где су
l
1
{\displaystyle l_{1}}
l
2
{\displaystyle l_{2}}
l
3
{\displaystyle l_{3}}
Претпоставимо да су два јединична вектора
x
{\displaystyle \mathbf {x} }
x
′
{\displaystyle \mathbf {x} '}
(
ϑ
,
φ
)
{\displaystyle (\vartheta ,\,\varphi )}
(
ϑ
′
,
φ
′
)
{\displaystyle (\vartheta ',\,\varphi ')}
cos
γ
=
cos
ϑ
cos
ϑ
′
+
sin
ϑ
sin
ϑ
′
cos
(
φ
−
φ
′
)
.
{\displaystyle \cos \gamma =\cos \vartheta \cos \vartheta '+\sin \vartheta \sin \vartheta '\cos(\varphi -\varphi ')\,.}
Адиционa теоремa за сферне хармонике је:
P
l
(
cos
γ
)
=
4
π
2
l
+
1
∑
m
=
−
l
l
Y
l
m
(
ϑ
,
φ
)
Y
l
m
∗
(
ϑ
′
,
φ
′
)
.
{\displaystyle P_{l}(\cos \gamma )={\frac {4\pi }{2l+1}}\sum _{m=-l}^{l}Y_{lm}(\vartheta ,\varphi )Y_{lm}^{*}(\vartheta ',\varphi ').}
За случај када се ради о истом вектору добија се:
∑
m
=
−
ℓ
ℓ
Y
ℓ
m
∗
(
θ
,
φ
)
Y
ℓ
m
(
θ
,
φ
)
=
2
ℓ
+
1
4
π
{\displaystyle \sum _{m=-\ell }^{\ell }Y_{\ell m}^{*}(\theta ,\varphi )\,Y_{\ell m}(\theta ,\varphi )={\frac {2\ell +1}{4\pi }}}
уреди
Пошто сферни хармоници чине потпун скуп опртонормалних функција функције могу да се развију преко њих:
f
(
θ
,
φ
)
=
∑
ℓ
=
0
∞
∑
m
=
−
ℓ
ℓ
f
ℓ
m
Y
ℓ
m
(
θ
,
φ
)
.
{\displaystyle f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}\,Y_{\ell }^{m}(\theta ,\varphi ).}
а коефицијенти су:
f
ℓ
m
=
∫
Ω
f
(
θ
,
φ
)
Y
ℓ
m
∗
(
θ
,
φ
)
d
Ω
=
∫
0
2
π
d
φ
∫
0
π
d
θ
sin
θ
f
(
θ
,
φ
)
Y
ℓ
m
∗
(
θ
,
φ
)
.
{\displaystyle f_{\ell }^{m}=\int _{\Omega }f(\theta ,\varphi )\,Y_{\ell }^{m*}(\theta ,\varphi )\,d\Omega =\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }\,d\theta \,\sin \theta f(\theta ,\varphi )Y_{\ell }^{m*}(\theta ,\varphi ).}
уреди
Првих неколико сферних хармоника
Ylm
l = 0
l = 1
l = 2
l = 3
m = -3
35
64
π
sin
3
ϑ
e
−
3
i
φ
{\displaystyle {\sqrt {\tfrac {35}{64\pi }}}\sin ^{3}{\vartheta }\,e^{-3i\varphi }}
m = −2
15
32
π
sin
2
ϑ
e
−
2
i
φ
{\displaystyle {\sqrt {\tfrac {15}{32\pi }}}\sin ^{2}{\vartheta }\,e^{-2i\varphi }}
105
32
π
sin
2
ϑ
cos
ϑ
e
−
2
i
φ
{\displaystyle {\sqrt {\tfrac {105}{32\pi }}}\sin ^{2}{\vartheta }\cos {\vartheta }\,e^{-2i\varphi }}
m = −1
3
8
π
sin
ϑ
e
−
i
φ
{\displaystyle {\sqrt {\tfrac {3}{8\pi }}}\sin {\vartheta }\,e^{-i\varphi }}
15
8
π
sin
ϑ
cos
ϑ
e
−
i
φ
{\displaystyle {\sqrt {\tfrac {15}{8\pi }}}\sin {\vartheta }\,\cos {\vartheta }\,e^{-i\varphi }}
21
64
π
sin
ϑ
(
5
cos
2
ϑ
−
1
)
e
−
i
φ
{\displaystyle {\sqrt {\tfrac {21}{64\pi }}}\sin {\vartheta }\left(5\cos ^{2}{\vartheta }-1\right)\,e^{-i\varphi }}
m = 0
1
4
π
{\displaystyle {\sqrt {\tfrac {1}{4\pi }}}}
3
4
π
cos
ϑ
{\displaystyle {\sqrt {\tfrac {3}{4\pi }}}\cos {\vartheta }}
5
16
π
(
3
cos
2
ϑ
−
1
)
{\displaystyle {\sqrt {\tfrac {5}{16\pi }}}\left(3\cos ^{2}{\vartheta }-1\right)}
7
16
π
(
5
cos
3
ϑ
−
3
cos
ϑ
)
{\displaystyle {\sqrt {\tfrac {7}{16\pi }}}\left(5\cos ^{3}{\vartheta }-3\cos {\vartheta }\right)}
m = 1
−
3
8
π
sin
ϑ
e
i
φ
{\displaystyle -{\sqrt {\tfrac {3}{8\pi }}}\sin {\vartheta }\,e^{i\varphi }}
−
15
8
π
sin
ϑ
cos
ϑ
e
i
φ
{\displaystyle -{\sqrt {\tfrac {15}{8\pi }}}\sin {\vartheta }\,\cos {\vartheta }\,e^{i\varphi }}
−
21
64
π
sin
ϑ
(
5
cos
2
ϑ
−
1
)
e
i
φ
{\displaystyle -{\sqrt {\tfrac {21}{64\pi }}}\sin {\vartheta }\left(5\cos ^{2}{\vartheta }-1\right)\,e^{i\varphi }}
m = 2
15
32
π
sin
2
ϑ
e
2
i
φ
{\displaystyle {\sqrt {\tfrac {15}{32\pi }}}\sin ^{2}{\vartheta }\,e^{2i\varphi }}
105
32
π
sin
2
ϑ
cos
ϑ
e
2
i
φ
{\displaystyle {\sqrt {\tfrac {105}{32\pi }}}\sin ^{2}{\vartheta }\cos {\vartheta }\,e^{2i\varphi }}
m = 3
−
35
64
π
sin
3
ϑ
e
3
i
φ
{\displaystyle -{\sqrt {\tfrac {35}{64\pi }}}\sin ^{3}{\vartheta }\,e^{3i\varphi }}
Abramowitz, Milton; Stegun, Irene A., eds. (1965), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover. ISBN 978-0-486-61272-0 .
Сферни хармоници Courant, Richard; Hilbert, David (1962), Methods of Mathematical Physics, Volume I , Wiley-Interscience Edmonds, A.R. (1957). Angular Momentum in Quantum Mechanics . Princeton University Press. ISBN 0-691-07912-9 . Eremenko, Alexandre; Jakobson, Dmitry; Nadirashvili, Nikolai (2007), „On nodal sets and nodal domains on
S
2
{\displaystyle S^{2}}
R
2
{\displaystyle \mathbb {R} ^{2}}
Université de Grenoble. Annales de l'Institut Fourier , 57 (7): 2345—2360, ISSN 0373-0956 , MR 2394544 MacRobert, T.M. (1967), Spherical harmonics: An elementary treatise on harmonic functions, with applications , Pergamon Press Meijer, Paul Herman Ernst; Bauer, Edmond (2004). Group theory: The application to quantum mechanics . Dover. ISBN 978-0-486-43798-9 . Solomentsev, E.D. (2001). „Spherical harmonics”. Ур.: Hazewinkel Michiel. Encyclopaedia of Mathematics ISBN 978-1556080104 . Stein, Elias; Weiss, Guido (1971). Introduction to Fourier Analysis on Euclidean Spaces ISBN 978-0-691-08078-9 . Unsöld, Albrecht (1927), „Beiträge zur Quantenmechanik der Atome”, Annalen der Physik , 387 (3): 355—393, Bibcode :1927AnP...387..355U , doi :10.1002/andp.19273870304 Watson, G. N.; Whittaker, E. T. (1927), A Course of Modern Analysis , Cambridge University Press, стр. 392
E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics , (1955) Chelsea Pub. Co. ISBN 978-0-8284-0104-3 .
C. Müller, Spherical Harmonics , (1966) Springer, Lecture Notes in Mathematics, Vol. 17. ISBN 978-3-540-03600-5 .
E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra , (1970) Cambridge at the University Press, ISBN 0-521-09209-4 , See chapter 3 .
J.D. Jackson, Classical Electrodynamics , ISBN 0-471-30932-X
Albert Messiah, Quantum Mechanics , volume II. (2000) Dover. ISBN 0-486-40924-4 .
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). „Section 6.7. Spherical Harmonics” . Numerical Recipes: The Art of Scientific Computing (3rd изд.). New York: Cambridge University Press. ISBN 978-0-521-88068-8 . D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii Quantum Theory of Angular Momentum ,(1988) World Scientific Publishing Co., Singapore, ISBN 9971-5-0107-4
![{\displaystyle \lambda \sin ^{2}(\theta )+{\frac {\sin(\theta )}{\Theta (\theta )}}{\frac {d}{d\theta }}\left[\sin(\theta ){\frac {d\Theta }{d\theta }}\right]=m^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6790186cfc51c7c9dd74e1345cda02e45285fd2d)
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