Sferni harmonici

Sferni harmonici u matematici predstavljaju ugaoni deo rešenja Laplasove jednačine u sfernim koordinatama.

Sferne harmonike je prvi 1782. uveo Pjer Simon Laplas, a oblika su:

$Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }$

i rešenje su jednačine:

${\frac {1}{\sin {\theta }}}{\frac {d}{d\theta }}\left(\sin {\theta }{\frac {dY_{\ell }^{m}}{d\theta }}\right)+(l(l+1)-{\frac {m^{2}}{\sin ^{2}{\theta }}})Y_{\ell }^{m}=0$

Laplasova jednačina

Laplasova jednačina u sfernim koordinatama ima oblik:

$\nabla ^{2}f={1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial f \over \partial r}\right)+{1 \over r^{2}\sin \theta }{\partial \over \partial \theta }\left(\sin \theta {\partial f \over \partial \theta }\right)+{1 \over r^{2}\sin ^{2}\theta }{\partial ^{2}f \over \partial \varphi ^{2}}=0.$

Jednačinu rešavamo separacijom varijabli pretpostavljajući rešenje oblika:

$f(r,\vartheta ,\varphi )=R(r)Y(\vartheta ,\varphi )$

Separacijom varijabli dobija se:

$\Delta R(r)Y(\vartheta ,\varphi )=Y(\vartheta ,\varphi )\Delta _{r}R(r)+{\frac {R(r)}{r^{2}}}\Delta _{\vartheta ,\varphi }Y(\vartheta ,\varphi )=0$

Množeći sa $r^{2}$ i deleći sa $R(r)Y(\vartheta ,\varphi )$ dobija se:

${\frac {r^{2}\Delta _{r}R(r)}{R(r)}}+{\frac {\Delta _{\vartheta ,\varphi }Y(\vartheta ,\varphi )}{Y(\vartheta ,\varphi )}}=0$

odnosno dobijaju se dve jednačine:

${\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)=\lambda ,\qquad {\frac {1}{Y}}{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial Y}{\partial \theta }}\right)+{\frac {1}{Y}}{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}Y}{\partial \varphi ^{2}}}=-\lambda .$

Ugaona jednačina

$\left({\frac {\partial ^{2}}{\partial \vartheta ^{2}}}+{\frac {\cos \vartheta }{\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}+{\frac {1}{\sin ^{2}\vartheta }}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}\right)Y(\vartheta ,\varphi )=-\lambda Y(\vartheta ,\varphi )$

može dalje da se separira po dve varijable:

$Y(\vartheta ,\varphi )=\Theta (\vartheta )\Phi (\varphi )$

Odatle se dobija:

$\underbrace {{\frac {\sin ^{2}\vartheta }{\Theta (\vartheta )}}\left({\frac {\partial ^{2}}{\partial \vartheta ^{2}}}+{\frac {\cos \vartheta }{\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}\right)\Theta (\vartheta )+\sin ^{2}(\vartheta )\lambda )} _{m^{2}}=\underbrace {-{\frac {1}{\Phi (\varphi )}}{\frac {\partial ^{2}}{\partial \varphi ^{2}}}\Phi (\varphi )} _{m^{2}}$

tj. dve jednačine:

${\frac {1}{\Phi (\varphi )}}{\frac {d^{2}\Phi (\varphi )}{d\varphi ^{2}}}=-m^{2}$

$\lambda \sin ^{2}(\theta )+{\frac {\sin(\theta )}{\Theta (\theta )}}{\frac {d}{d\theta }}\left[\sin(\theta ){\frac {d\Theta }{d\theta }}\right]=m^{2}$

Rešenje prve jednačine je: $\Phi _{m}(\varphi )=A\exp(im\varphi )$

Da bi druga jednačina imala rešenje mora biti zadovoljeno $\lambda =l(l+1)$.

Konačno za ugao $\theta$ dobija se jednačina:

${\frac {1}{\sin {\theta }}}{\frac {d}{d\theta }}\left(\sin {\theta }{\frac {d\Theta _{lm}}{d\theta }}\right)-{\frac {m^{2}}{\sin ^{2}{\theta }}}\Theta _{lm}+l(l+1)\Theta _{lm}=0$

Uvedemo li supstituciju $x=\cos(\theta )$ dobija se:

$(1-x^{2})\,\Theta ''-2x\Theta '+\left(\ell [\ell +1]-{\frac {m^{2}}{1-x^{2}}}\right)\,\Theta =0,\,$<

odnosno jednačina čije rešenje su pridruženi Ležandrovi polinomi $P_{lm}(\cos \vartheta )$. Sada treba da normiramo ta rešenja uz pomoć $\int _{0}^{\pi }|\Theta _{lm}(\vartheta )|^{2}\sin(\vartheta )\mathrm {d} \vartheta =1$ pa dobijamo:

$\Theta _{lm}(\vartheta )={\sqrt {{\frac {2l+1}{2}}\cdot {\frac {(l-m)!}{(l+m)!}}}}\,\,P_{lm}(\cos \vartheta )$

Isto tako treba da se normira i po drugom uglu $\int _{0}^{2\pi }|\Phi _{m}(\varphi )|^{2}\mathrm {d} \varphi =1$ , pa se dobija:

$\Phi _{m}(\varphi )={\frac {1}{\sqrt {2\pi }}}\exp(im\varphi ),\quad m\in \mathbb {Z}$.

Zajedničko ugaono rešenje je onda upravo funkcija, koju nazivamo sferni harmonik:

$Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }$

Neka svojstva

Sferni harmonici su ortogonalni:

$\int _{\theta =0}^{\pi }\int _{\varphi =0}^{2\pi }Y_{\ell }^{m}\,Y_{\ell '}^{m'*}\,d\Omega =\delta _{\ell \ell '}\,\delta _{mm'},$.

Zadovoljavaju relaciju potpunosti:

$\sum _{l=0}^{\infty }\sum _{m=-l}^{l}Y_{lm}^{*}(\vartheta ',\varphi ')\,Y_{lm}(\vartheta ,\varphi )=\delta (\varphi -\varphi ')\delta (\cos {\vartheta }-\cos {\vartheta '})$

Osim toga u slučaju transformacija vredi:

$Y_{lm}(\pi -\vartheta ,\pi +\varphi )=(-1)^{l}\cdot Y_{lm}(\vartheta ,\varphi )$

$Y_{l,-m}(\vartheta ,\varphi )=(-1)^{m}\cdot Y_{lm}^{*}(\vartheta ,\varphi )$

Integral tri sferna harmonika dat je preko 3-jm simbola:

${\begin{aligned}&{}\quad \int Y_{l_{1}m_{1}}(\theta ,\varphi )Y_{l_{2}m_{2}}(\theta ,\varphi )Y_{l_{3}m_{3}}(\theta ,\varphi )\,\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi \\&={\sqrt {\frac {(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi }}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\[8pt]0&0&0\end{pmatrix}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\end{aligned}}$

gde su $l_{1}$, $l_{2}$ and $l_{3}$ celi brojevi.

Adiciona teorema

Pretpostavimo da su dva jedinična vektora $\mathbf {x}$ i $\mathbf {x} '$ predstavljena u sfernim kordinatama $(\vartheta ,\,\varphi )$ odnosno $(\vartheta ',\,\varphi ')$. Ugao između dva vektora je onda:

$\cos \gamma =\cos \vartheta \cos \vartheta '+\sin \vartheta \sin \vartheta '\cos(\varphi -\varphi ')\,.$

Adiciona teorema za sferne harmonike je:

$P_{l}(\cos \gamma )={\frac {4\pi }{2l+1}}\sum _{m=-l}^{l}Y_{lm}(\vartheta ,\varphi )Y_{lm}^{*}(\vartheta ',\varphi ').$

Za slučaj kada se radi o istom vektoru dobija se:

$\sum _{m=-\ell }^{\ell }Y_{\ell m}^{*}(\theta ,\varphi )\,Y_{\ell m}(\theta ,\varphi )={\frac {2\ell +1}{4\pi }}$

Razvoj po sfernim harmonicima

Pošto sferni harmonici čine potpun skup oprtonormalnih funkcija funkcije mogu da se razviju preko njih:

$f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}\,Y_{\ell }^{m}(\theta ,\varphi ).$

a koeficijenti su:

$f_{\ell }^{m}=\int _{\Omega }f(\theta ,\varphi )\,Y_{\ell }^{m*}(\theta ,\varphi )\,d\Omega =\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }\,d\theta \,\sin \theta f(\theta ,\varphi )Y_{\ell }^{m*}(\theta ,\varphi ).$

Tabela nekih sfernih harmonika

Prvih nekoliko sfernih harmonika

Ylm	l = 0	l = 1	l = 2	l = 3
m = -3				${\sqrt {\tfrac {35}{64\pi }}}\sin ^{3}{\vartheta }\,e^{-3i\varphi }$
m = −2			${\sqrt {\tfrac {15}{32\pi }}}\sin ^{2}{\vartheta }\,e^{-2i\varphi }$	${\sqrt {\tfrac {105}{32\pi }}}\sin ^{2}{\vartheta }\cos {\vartheta }\,e^{-2i\varphi }$
m = −1		${\sqrt {\tfrac {3}{8\pi }}}\sin {\vartheta }\,e^{-i\varphi }$	${\sqrt {\tfrac {15}{8\pi }}}\sin {\vartheta }\,\cos {\vartheta }\,e^{-i\varphi }$	${\sqrt {\tfrac {21}{64\pi }}}\sin {\vartheta }\left(5\cos ^{2}{\vartheta }-1\right)\,e^{-i\varphi }$
m = 0	${\sqrt {\tfrac {1}{4\pi }}}$	${\sqrt {\tfrac {3}{4\pi }}}\cos {\vartheta }$	${\sqrt {\tfrac {5}{16\pi }}}\left(3\cos ^{2}{\vartheta }-1\right)$	${\sqrt {\tfrac {7}{16\pi }}}\left(5\cos ^{3}{\vartheta }-3\cos {\vartheta }\right)$
m = 1		$-{\sqrt {\tfrac {3}{8\pi }}}\sin {\vartheta }\,e^{i\varphi }$	$-{\sqrt {\tfrac {15}{8\pi }}}\sin {\vartheta }\,\cos {\vartheta }\,e^{i\varphi }$	$-{\sqrt {\tfrac {21}{64\pi }}}\sin {\vartheta }\left(5\cos ^{2}{\vartheta }-1\right)\,e^{i\varphi }$
m = 2			${\sqrt {\tfrac {15}{32\pi }}}\sin ^{2}{\vartheta }\,e^{2i\varphi }$	${\sqrt {\tfrac {105}{32\pi }}}\sin ^{2}{\vartheta }\cos {\vartheta }\,e^{2i\varphi }$
m = 3				$-{\sqrt {\tfrac {35}{64\pi }}}\sin ^{3}{\vartheta }\,e^{3i\varphi }$

Literatura

• 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 $S^{2}$ and $\mathbb {R} ^{2}$”, Université de Grenoble. Annales de l'Institut Fourier, 57 (7): 2345—2360, ISSN 0373-0956, MR2394544 
• 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. Springer. ISBN 978-1556080104. .
• Stein, Elias; Weiss, Guido (1971). Introduction to Fourier Analysis on Euclidean Spaces. Princeton, N.J.: Princeton University Press. 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, str. 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