% qm2.tex Quantenmechanik WS 1984 / 85 % % Y - Formelsammlung % def.tex erforderlich %%%% T E X %%%% % % \magnification=\magstep1 \nopagenumbers \voffset=-2.2cm \vsize=28.4cm \hoffset=.5cm \hsize=17cm \font\gr=cmbx12 scaled \magstep1 \font\grr=cmsl10 scaled \magstep4 \font\mt=cmsl9 \font\kl=cmbx6 % \baselineskip 14pt \input def \def\eq#1{\eqno{(\, {\rm Y} \, . \, #1 \, )}} \def\gl#1{\hbox{$\, (\, {\rm Y} \, . \, $#1$\, )\;$}} \def\sz{\vskip -.04cm \nz} \def\halb{\nz \hskip -2cm \vrule depth 0pt height .05pt width 4cm \vskip .1cm \nz} \def\ganz{\sz \hskip -2cm \vrule depth 0pt height 2pt width 19cm \medskip \nz} \phantom{o} \nz {Q. , \ WS 1984/85 \hfill \hfill {\grr Y}} \nz \ct {\qquad $Y_{\ell\, m}$ , \ $j_{\ell}$ , \ $\ph _{n\, \ell\, m}$ \quad \vrule depth -1.6pt height 2.6pt width .3cm \quad {\bf Formelsammlung}} \bigskip \nz Die Quantenmechanik ist leider reich an technischen Details, deren Erarbeitung zun\"achst nur wenig Gewinn abwirft, wohl aber dann deren Einbau in physikalische Strukturen. Einmal im Leben (verifizieren) --- und fortan von der Erinnerung zehren (z.B. von dieser hier)$\,$. \smallskip \nz Die {\bf Kugel}fl\"achen{\bf funktionen} (salopp \anfu Kugelfunktionen\anfo , spherical harmonics) sind per def. die simultanen Eigenfunktionen zweier miteinander vertauschbarer, nur auf Kugelwinkel wirkender, hermitescher Operatoren, n\"amlich von \smallskip \nz $$ \hbox{$ \vc L^2 = (\vc r \times \vc p )^2 = - \hbar^2 r^2 \( \D - \D_r \) = - \hbar^2 \lk {1 \0 \sin (\ta ) } \,\6_\ta \sin (\ta ) \,\6_\ta + {1\0 \sin^2 (\ta )} \6_\ph^2 \rk \;\; {\rm und} \;\;\; L_z = {\hbar \0 i} \6_\ph $} \eq 1 $$ \nz zu Eigenwerten $\;\; \hbar^2 \ell (\ell + 1)\;\;$ bzw. $\;\;\hbar m\;\;$, wobei $\;\;\ell = 0,1,2, \ldots\;\;$ und $\;\; m=-\ell, -\ell +1, \ldots , \ell-1 , \ell\;$. \nz Diese $Y_{\ell\, m} ( \ta, \ph )$ (oder $Y_{\ell\, m} (\Omega )$) sind folglich othonormierbar und bilden ein VONS ({\bf V}ollst\"andiges {\bf O}rtho\-{\bf N}ormal\-{\bf S}ystem) im Raum der Funktionen $f(\ta, \ph)$ \"uber der Einheitskugel$\,$. \halb $$ \int \! d\Omega \,\; Y_{\ell\, m}^* \; Y_{\ell^\prime \, m^\prime} = \d_{\ell \, \ell^\prime} \,\d_{m\, m^\prime} \qquad , \qquad Y_{\ell\, m} ( \ta, \ph ) = \a_{\ell \, m} \, P_\ell^{\, \vert m \vert } \( \cos (\ta ) \) \, e^{im\ph} \eq 2 $$ \sz $$ P_\ell^{\, \vert m \vert } (x) = (1-x^2)^{\vert m \vert /2} \,\6_x^{\vert m \vert} P_\ell (x) \qquad , \qquad P_\ell (x) = {1\0 2^\ell \, \ell ! } \6_x^\ell \, (x^2 - 1 )^\ell \qquad\quad \eq 3 $$ \sz \hskip 2.7cm zugeordnete Legendre--\anfu Polynome\anfo \hskip 1.4cm Legendre--Polynome \vskip .2cm \nz $$ \a_{\ell\, m} = \wu { (2\ell + 1) / 4\pi } \,\wu { (\ell - \vert m \vert ) !\, /\, (\ell + \vert m \vert ) ! } \( - m / \vert m \vert \)^m \qquad \( 0^0 \gll 1 \) \eq 4 $$ \vskip -.3cm \sz \halb $$ Y_{0 \, 0 } = {1 \0 \wu {4\pi } } \quad ; \quad Y_{1 \, 0 } = \wu {{3\0 4\pi }} \cos (\ta ) \quad , \quad Y_{1 \, \pm 1 } = \mp \wu {{3\0 8\pi }} \sin (\ta ) \, e^{\pm i \ph } \quad ; \eq 5 $$ \sz $$ Y_{2 \, 0 } = \wu {{5\0 16\pi }} \( 3\cos^2 (\ta ) -1\) \;\; , \;\; Y_{2 \, \pm 1 } = \mp \wu {{15\0 8\pi }} \sin (\ta ) \cos (\ta ) e^{\pm i\ph } \;\; , \;\; Y_{2 \, \pm 2 } = \wu {{15\0 32\pi }} \sin^2 (\ta ) e^{\pm i 2 \ph } \quad . \eq 6 $$ \sz \halb \vskip .1cm \nz Erzeugende Funktion$\,$: \vskip -1.2cm $$ \hskip 3.7cm \wu {{1\0 1 - 2ux + u^2 }} = \sum_{\ell =0}^\infty u^\ell P_\ell (x) \;\; , \;\; {1\0 \vert \vc r - \vc r^\prime \vert } = \sum_{\ell =0}^\infty { (r^\prime )^\ell \0 r^{\ell +1} } P_\ell \(\cos (\ta ) \) \;\; \hbox{f\"ur}\; r^\prime < r \eq 7 $$ \nz Additionstheorem$\,$: \vskip -.95cm $$ \hskip .4cm Y_{\ell\, 0} \( \ta \gll \hbox{{\fiverm Winkel zwischen}} \,\vc r, \vc r^\prime \) \; = \;\wu {{4\pi \0 2\ell +1}} \sum_{m=-\ell}^\ell \; Y_{\ell\, m} ( \Omega ) \,\; Y_{\ell\, m}^* (\Omega^\prime ) \hfill \eq 8 $$ \sz $$ L_\pm \gll L_x \pm i L_y = \hbar\, e^{\pm i \ph } \( \pm \6_\ta + i\, {\rm ctg} (\ta ) \,\6_\ph \) \quad , \quad \vc L^2 = L_+ L_- + L_z^2 - \hbar L_z \quad ; \eq 9 $$ \sz $$ L_\pm Y_{\ell\, m} = \hbar \wu {\ell (\ell + 1 ) - m (m\pm 1) } Y_{\ell \, m \pm 1 } \quad , \quad Y_{\ell\,\ell} = {(-1)^\ell \0 2^\ell \, \ell ! } \wu {{ (2\ell +1) ! \0 4\pi }} \sin^\ell (\ta ) \; e^{i\,\ell\,\ph } \quad . \eq {10} $$ \sz %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \ganz {\bf Sph\"arische Besselfunktionen} \ \ l\"osen den Radialanteil der station\"aren Schr\"odinger--Gleichung im potentialfreien (meist Au\ss en--)Raum \nz $$ {\hbar^2 \0 2m} \( - {1\0 r} \6_r^2 r + {\ell (\ell +1) \0 r^2} \) R(r) = {\hbar^2 k^2 \0 2m} R(r) \quad , \quad R(r) = A\, j_\ell (kr) + B \, n_\ell (kr) \quad , \quad \rho \gll kr \eq {11} $$ \sz $$ \Rightarrow \qquad \( - {1\0 \rho} \6_\rho^2 \rho + {\ell (\ell +1) \0 \rho^2} - 1 \) \lb \matrix{ j_\ell (\rho ) \cr n_\ell (\rho ) \cr } \rb = 0 \qquad\; \( \; \lower 5pt\vbox{\hbox{\fiverm Zusammenhang} \vskip -.2cm \hbox{\fiverm mit halbzahligen} \vskip -.2cm \hbox{\fiverm Bessel--Funktionen : } } \;\; \matrix{ j_\ell = \wu {\pi / 2\rho } J_{\ell + {1\02}} \cr n_\ell = \wu {\pi / 2\rho } Y_{\ell + {1\02}} \cr } \; \) \eq{12} $$ \sz \halb $$ j_\ell (\rho ) = (-\rho )^\ell \( {1\0 \rho} \6_\rho \)^\ell {\sin (\rho ) \0 \rho } \; \to \; \lb \matrix{ \rho^\ell / ( 2\ell + 1 ) !! \quad (\rho \to 0 ) \cr {1\0 \rho} \sin \( \rho - \ell {\pi\02} \) \quad (\rho \to \infty ) \cr} \right. \eq{13} $$ \sz $$ n_\ell (\rho ) = - (-\rho )^\ell \( {1\0 \rho } \6_\rho \)^\ell {\cos (\rho ) \0 \rho } \; \to \; \lb \matrix{ - ( 2\ell - 1 ) !! \,\rho^{-\ell -1} \quad (\rho \to 0 ) \cr - {1\0 \rho} \cos \( \rho - \ell {\pi\02} \) \quad (\rho \to \infty ) \cr} \right. \eq{14} $$ \sz $$ h_\ell^{(1)} (\rho ) \gll j_\ell (\rho ) + i\, n_\ell (\rho ) \qquad \to \quad {1 \0 i \rho} e^{i \(\rho - \ell {\pi\02} \)} \quad ; \quad\; E < 0 \; : \;\; k \to i\kappa \; , \; i\rho \to - \kappa r \;\; . \eq{15} $$ \sz \halb \vskip -.2cm \nz $$ e^{ikz} = e^{ikr \cos (\ta )} = \sum_{\ell =0}^\infty ( 2\ell +1) \, i^\ell \, j_\ell (kr) \, P_\ell \(\cos (\ta ) \) \eq{16} $$ \sz %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \ganz {\bf H--Atom--Energie--Eigenzust\"ande} \vskip -.45cm \nz $$ \hskip 6.5cm \ph_{n\,\ell \, m} ( r,\ta , \ph ) = R_{n\,\ell} (r) \; Y_{\ell\, m } (\ta , \ph ) \eq{17} $$ \sz $$ \int_0^\infty \! dr \; r^2 \, R^2_{n\, \ell} (r) = 1 \quad , \quad R_{n\,\ell} (r) = \( \sum_{\nu = 0}^{n-\ell -1} c_\nu \rho^\nu \) \rho^\ell e^{-\rho / n} \;\; , \;\; \rho = {r\0 a} \;\; , \;\; a = {\hbar^2 \0 \mu e_0^2 } \;\; , \;\; e_0^2 \gll {e^2 \0 4\pi \e_0 } \; \eq{18} $$ \sz \vskip -.2cm \nz $$ c_{\nu + 1} = {2\0 n} \; {\nu + \ell + 1 -n \0 (\nu + 2\ell + 2) \, ( \nu +1) } \; c_\nu \;\; ; \;\; n=1,2, \ldots \;\; ; \;\; E = - \( {e_0^2 \0 2 a } \) \, {1 \0 n^2} \;\; ; \;\; n^\prime = n - \ell -1 = \; \lower 5pt\vbox{\hbox{\fiverm Zahl der} \vskip -.2cm \hbox{\fiverm radialen} \vskip -.2cm \hbox{\fiverm Nullstellen}} \eq{19} $$ \sz \halb \vskip .1cm \nz $$ \matrix{ R_{1\, 0} = a^{-3/2} 2 e^{-\rho} \hfill & n^\prime = 0 \hskip 1cm & R_{3\, 0} = a^{-3/2} {2\0 3\wu 3 } \( 1 - {2\03} \rho + {2\0 27} \rho^2 \) e^{-\rho / 3} \hfill & n^\prime = 2 \cr % R_{2\, 0} = a^{-3/2} {1\0 \wu 2 } \( 1- {1\02} \rho \) e^{-\rho / 2} \hfill & n^\prime = 1 \hfill & R_{3\, 1} = a^{-3/2} {8\0 27 \wu 6 } \rho \( 1-{1\06} \rho \) e^{-\rho / 3} \hfill & n^\prime = 1 \cr % R_{2\, 1} = a^{-3/2} {1\0 2 \wu 6 } \rho e^{-\rho / 2} \hfill & n^\prime = 0 \hfill & R_{3\, 2} = a^{-3/2} {4\0 81 \wu {30} } \rho^2 e^{-\rho / 3 } \hfill & n^\prime = 0 \cr } \eq{20} $$ \halb \nz {\fiverm ( hschulz@itp.uni-hannover.de ) } \end