
%  qm2.tex        Quantenmechanik  WS 1984 / 85
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\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
