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{\bf Pou\v{z}itie varia\v{c}n\'{e}ho princ\'{i}pu. } 
Vypracova\q{t} do 11.4.2000, ot\'{a}zky: bokes@elf.stuba.sk \\

{\it \v{C}as\q{t} 1.}
      Na z\'{a}klade anal\'{y}zy radi\'{a}lnej rovnice pre elektr\'{o}n
      v at\'{o}me vod\'{i}ka 
\begin{equation}
       \label{SchR}
      -\frac{1}{2}\frac{1}{r^{2}}\frac{\partial}{\partial r }
       r^{2}\frac{\partial}{\partial r} u(r) - \frac{1}{\rho}
      + \frac{ l(l+1)}{2 r^{2} }u( r ) = \epsilon u(r) 
\end{equation}
v dvoch limitn\'{y}ch pr\'{i}padoch $r \rightarrow 0$ a $r \rightarrow 
\infty$,
kde $r \rightarrow \frac{ \hbar^{2} }{m e^{2}} r $ a $E = \frac{m e^{4}}{\hbar^{2}}
\epsilon $, sa d\'{a} uk\'{a}za\q{t}, \v{z}e:
\begin{equation}
	\label{func}
      u_{l}(r) \sim r^{l}(a_{0} + a_{1}r + a_{2}r^{2} + .... )
                {\rm e}^{-\sqrt{|\epsilon|}r}.
\end{equation}
Pou\v{z}it\'{i}m varia\v{c}n\'{e}ho princ\'{i}pu spo\v{c}\'{i}tajte
presn\'{u} energiu stavu n=1,l=0 ( n=1 zodpoved\'{a} najjednoduch\v{s}iemu
polyn\'{o}mu, t.j. $a_{0}= const$, $a_{i}=0$ pre i=1,2,3,... ), pri\v{c}om
$\sqrt{|\epsilon|}$ vo (\ref{func}) pova\v{z}ujte za varia\v{c}n\'{y} 
parameter $\alpha$. \\
Pom\^{o}cky:
\begin{itemize}
\item pre l=0,n=1 je $u(r) \sim a_{0}{\rm e}^{-\alpha r}$, normovan\'{i}m
tejto funkcii ur\v{c}te $a_{0}$. 
\item spo\v{c}\'{i}tajte $\langle \alpha | H | \alpha \rangle$ pre 
lubovoln\'{e}$\alpha$ ( $ | \alpha \rangle $ je u(r); spr\'{a}vny v\'{y}sledok 
je $ \langle \alpha | H | \alpha \rangle = \alpha( \frac{\alpha}{2} -1 ) $).
 N\'{a}jdite minimum posledn\'{e}ho v\'{y}razu. Hamiltoni\'{a}n 
je dan\'{y} \q{l}avou stranou (\ref{SchR}). Pozor, $ \langle \alpha | H | 
\alpha \rangle =$ \- $ \int_{0}^{\infty} dr r^{2} u(r) H u(r) $, kde $r^{2}$ 
poch\'{a}dza z Jakobi\'{a}nu pri prechode $(x,y,z) \rightarrow 
(r,\phi,\theta)$ pod integr\'{a}lom.
Uhlov\'{e} premenn\'{e} s\'{u} vyintegrovan\'{e}, a to tak, \v{z}e po nich 
ni\v{c} nezostalo.

\item Dok\'{a}\v{z}te si dopredu, \v{z}e 
\begin{equation}
   \int_{0}^{\infty}r^{n}{\rm e}^{-ar}dr = \frac{ n !}{a^{n+1}}.
\end{equation}
Vyu\v{z}ijete to ve\q{l}a kr\'{a}t.
\end{itemize}

Na premyslenie a \q{d}al\v{s}\'{i} rozvoj ducha:
\begin{itemize}
\item Overte si, \v{z}e rovnica (\ref{SchR}) s\'{u}hlas\'{i} zo 
Schr\"{o}dingerovou rovnicou pre jeden elektr\'{o}n v s\'{u}radniciach 
($r,\phi,\theta$) s potenci\'{a}lom 
$V(r) = \frac{ - e^{2} }{r}$.
Pou\v{z}it\'{a} z\'{a}mena jednotiek energii a vzdialenosti renderuje 
\'{u}lohu do jej prirodzen\'{y}ch rozmerov; E $\sim$ eV, r $\sim$ 0.1 nm. 
\item Ako by ste uk\'{a}zali, \v{z}e n\'{a}jden\'{y} stav $u_{\alpha_{min}}(r)$
je presn\'{y}m vlastn\'{y}m stavom, nie len optimalizovan\'{y}m pomocou 
varia\v{c}n\'{e}ho princ\'{i}pu?
\end{itemize}
 
{\it \v{C}as\q{t} 2.}
Spravte to ist\'{e}, ale s funkciou 
$| \alpha \rangle = A{\rm e}^{-\alpha r^{2}} $.
Tak\'{y}to ( Gaussi\'{a}n ) tvar sa ve\q{l}mi \v{c}asto pou\v{z}\'{i}va v 
numerick\'{y}ch v\'{y}po\v{c}toch na molekul\'{a}ch. Ur\v{c}te chybu
energii oproti presn\'{e}mu rie\v{s}eniu. [V\'{y}sledok je 
$\langle \alpha | H | \alpha \rangle = 
	\frac{3}{2}\alpha - 2 \sqrt{\frac{2 \alpha }{\pi} }; 
        \alpha_{min}\approx 0.283;
        E_{alpha_{min}}\approx - 0.424 
$]

Pom\^{o}cky:
\begin{itemize}
\item $\int_{0}^{\infty} {\rm e}^{-ar^{2}}dr = \sqrt{\frac{\pi}{4a}}$ 
      Dokazuje sa to pomocou \\
\begin{equation}
\left( \int_{-\infty}^{\infty}dr{\rm e}^{-r^{2}} \right)^{2} = 
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dxdy{\rm e}^{-x^{2}-y^{2}} = 
\int_{0}^{\infty}dr\int_{0}^{2 \pi} d \phi {\rm e}^{-r^{2}}r = ... 
\end{equation}
\item Dok\'{a}\v{z}te si: ( $n!! = n(n-2)(n-4)...$ ) \\
\begin{eqnarray}
\int_{0}^{\infty} dr r^{n} {\rm e}^{-ar^{2}} = \left\{ \begin{array}{ll}
               (n-1)!!\frac{\sqrt{\pi /2 }}{(2a)^{(n+1)/2}} & n\textrm{ p\'{a}rne }\\
               (n-1)!!\frac{1}{(2a)^{(n+1)/2}} & n\textrm{ nep\'{a}rne}  
               \end{array}
\right.
\end{eqnarray}
\item ostatn\'{e} je analogick\'{e} predch\'{a}dzaj\'{u}cemu pr\'{i}kladu.
\end{itemize}

{\it \v{C}as\q{t} 3.}

Na z\'{a}klade v\v{s}eobecn\'{e}ho tvaru maticov\'{y}ch elementov pre
oper\'{a}tory momentu hybnosti ${\rm \langle jm|L_{z}|j'm'\rangle}$, ${\rm 
\langle jm|L^{\pm}|j'm'\rangle}$, ${\rm L_{x}=\frac{1}{2}(L^{+} + L^{-}), L_{y}
}$ 
podobne, ur\v{c}te tvar oper\'{a}tora ( 2x2 matica ) spinu jedn\'{e}ho 
elektr\'{o}nu $s^{\alpha}, \alpha = x,y,z$. Ak je eletr\'{o}n vo vlastnom 
stave $s^{z}$ s vlastnou hodnotou $\frac{1}{2}$, ak\'{a} je 
pravdepodobnos\q{t}, \v{z}e priemet jeho spinu na os {\it y} je tie\v{z} 
$\frac{1}{2}$? \\

Pom\^{o}cka: Spinov\'{e} oper\'{a}tory $s^{\alpha}$ s\'{u} 2x2 matice, a ich 
vlastn\'{e} stavy s\'{u} 2-dimenzion\'{a}lne vektory (~oby\v{c}ajn\'{e}
vlastn\'{e} vektory z algebry ). 


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