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Next: 7. Verteilungsfunktionen Up: 6. Quantenmechanik Previous: 6.2 Winkelanteil der Wellenfunktionen   Inhalt

6.3 Normierte Eigenfunktionen $ \psi_{n,l,m}^{}$(r,$ \theta$,$ \phi$)

$\displaystyle \psi_{100}^{}$ = $\displaystyle {1 \over {\sqrt{\pi}}}$$\displaystyle \left(\vphantom{ {Z \over {a_0}}}\right.$$\displaystyle {Z \over {a_0}}$ $\displaystyle \left.\vphantom{ {Z \over {a_0}}}\right)^{3 \over 2 }_{}$exp(- $\displaystyle {{Zr}\over {a_0}}$) = R10 . Y00
$\displaystyle \psi_{200}^{}$ = $\displaystyle {1\over {4 \sqrt{2 \pi}}}$$\displaystyle \left(\vphantom{ {Z \over {a_0}}}\right.$$\displaystyle {Z \over {a_0}}$ $\displaystyle \left.\vphantom{ {Z \over {a_0}}}\right)^{3 \over 2 }_{}$$\displaystyle \left(\vphantom{ 2-{{Zr}\over {a_0}}}\right.$2 - $\displaystyle {{Zr}\over {a_0}}$ $\displaystyle \left.\vphantom{ 2-{{Zr}\over {a_0}}}\right)$exp(- $\displaystyle {{Zr}\over {2 a_0}}$) = R20 . Y00
$\displaystyle \psi_{210}^{}$ = $\displaystyle {1\over {4 \sqrt{2 \pi}}}$$\displaystyle \left(\vphantom{ {Z \over {a_0}}}\right.$$\displaystyle {Z \over {a_0}}$ $\displaystyle \left.\vphantom{ {Z \over {a_0}}}\right)^{3 \over 2 }_{}$$\displaystyle \left(\vphantom{ {{Zr}\over {a_0}}}\right.$$\displaystyle {{Zr}\over {a_0}}$ $\displaystyle \left.\vphantom{ {{Zr}\over {a_0}}}\right)$exp(- $\displaystyle {{Zr}\over {2 a_0}}$)cos($\displaystyle \theta$) = R21 . Y10
$\displaystyle \psi_{21\pm 1}^{}$ = $\displaystyle {1\over {8 \sqrt{\pi}}}$$\displaystyle \left(\vphantom{ {Z \over {a_0}}}\right.$$\displaystyle {Z \over {a_0}}$ $\displaystyle \left.\vphantom{ {Z \over {a_0}}}\right)^{3 \over 2 }_{}$$\displaystyle \left(\vphantom{ {{Zr}\over {a_0}}}\right.$$\displaystyle {{Zr}\over {a_0}}$ $\displaystyle \left.\vphantom{ {{Zr}\over {a_0}}}\right)$exp(- $\displaystyle {{Zr}\over {2 a_0}}$)sin($\displaystyle \theta$)e($\scriptstyle \pm$i$\scriptstyle \phi$) = R21 . Y1$\scriptstyle \pm$1
$\displaystyle \psi_{300}^{}$ = $\displaystyle {1\over {81 \sqrt{3 \pi}}}$$\displaystyle \left(\vphantom{ {Z \over {a_0}}}\right.$$\displaystyle {Z \over {a_0}}$ $\displaystyle \left.\vphantom{ {Z \over {a_0}}}\right)^{3 \over 2 }_{}$$\displaystyle \left(\vphantom{27 - 18 {{Zr}\over {a_0}} + 2 \left({{Zr}\over {a_0}} \right)^2}\right.$27 - 18$\displaystyle {{Zr}\over {a_0}}$ + 2$\displaystyle \left(\vphantom{ {{Zr}\over {a_0}}}\right.$$\displaystyle {{Zr}\over {a_0}}$ $\displaystyle \left.\vphantom{{{Zr}\over {a_0}} }\right)^{2}_{}$ $\displaystyle \left.\vphantom{27 - 18 {{Zr}\over {a_0}} + 2 \left({{Zr}\over {a_0}} \right)^2}\right)$exp(- $\displaystyle {{Zr}\over {3 a_0}}$) = R30 . Y00
$\displaystyle \psi_{310}^{}$ = $\displaystyle {{\sqrt{2}}\over {81 \sqrt{\pi}}}$$\displaystyle \left(\vphantom{ {Z \over {a_0}}}\right.$$\displaystyle {Z \over {a_0}}$ $\displaystyle \left.\vphantom{ {Z \over {a_0}}}\right)^{3 \over 2 }_{}$$\displaystyle \left(\vphantom{6 - {{Zr}\over {a_0}} }\right.$6 - $\displaystyle {{Zr}\over {a_0}}$ $\displaystyle \left.\vphantom{6 - {{Zr}\over {a_0}} }\right)$$\displaystyle {{Zr}\over {a_0}}$exp(- $\displaystyle {{Zr}\over {3 a_0}}$)cos($\displaystyle \theta$) = R31 . Y10
$\displaystyle \psi_{31\pm 1}^{}$ = $\displaystyle {1 \over {81 \sqrt{\pi}}}$$\displaystyle \left(\vphantom{ {Z \over {a_0}}}\right.$$\displaystyle {Z \over {a_0}}$ $\displaystyle \left.\vphantom{ {Z \over {a_0}}}\right)^{3 \over 2 }_{}$$\displaystyle \left(\vphantom{6 - {{Zr}\over {a_0}} }\right.$6 - $\displaystyle {{Zr}\over {a_0}}$ $\displaystyle \left.\vphantom{6 - {{Zr}\over {a_0}} }\right)$$\displaystyle {{Zr}\over {a_0}}$exp(- $\displaystyle {{Zr}\over {3 a_0}}$)sin($\displaystyle \theta$)e$\scriptstyle \pm$i$\scriptstyle \phi$ = R31 . Y1$\scriptstyle \pm$1
$\displaystyle \psi_{320}^{}$ = $\displaystyle {1 \over {81 \sqrt{6 \pi}}}$$\displaystyle \left(\vphantom{ {Z \over {a_0}}}\right.$$\displaystyle {Z \over {a_0}}$ $\displaystyle \left.\vphantom{ {Z \over {a_0}}}\right)^{3 \over 2 }_{}$$\displaystyle \left(\vphantom{ {{Zr}\over {a_0}}}\right.$$\displaystyle {{Zr}\over {a_0}}$ $\displaystyle \left.\vphantom{{{Zr}\over {a_0}} }\right)^{2}_{}$exp(- $\displaystyle {{Zr}\over {3 a_0}}$)(3cos2($\displaystyle \theta$) - 1) = R32 . Y20
$\displaystyle \psi_{32\pm 1}^{}$ = $\displaystyle {1 \over {81 \sqrt{\pi}}}$$\displaystyle \left(\vphantom{ {Z \over {a_0}}}\right.$$\displaystyle {Z \over {a_0}}$ $\displaystyle \left.\vphantom{ {Z \over {a_0}}}\right)^{3 \over 2 }_{}$$\displaystyle \left(\vphantom{ {{Zr}\over {a_0}}}\right.$$\displaystyle {{Zr}\over {a_0}}$ $\displaystyle \left.\vphantom{{{Zr}\over {a_0}} }\right)^{2}_{}$exp(- $\displaystyle {{Zr}\over {3 a_0}}$)sin($\displaystyle \theta$)cos($\displaystyle \theta$) - 1)e$\scriptstyle \pm$i$\scriptstyle \phi$ = R32 . Y2$\scriptstyle \pm$1
$\displaystyle \psi_{32\pm 2}^{}$ = $\displaystyle {1 \over {81 \sqrt{\pi}}}$$\displaystyle \left(\vphantom{ {Z \over {a_0}}}\right.$$\displaystyle {Z \over {a_0}}$ $\displaystyle \left.\vphantom{ {Z \over {a_0}}}\right)^{3 \over 2 }_{}$$\displaystyle \left(\vphantom{ {{Zr}\over {a_0}}}\right.$$\displaystyle {{Zr}\over {a_0}}$ $\displaystyle \left.\vphantom{{{Zr}\over {a_0}} }\right)^{2}_{}$exp(- $\displaystyle {{Zr}\over {3 a_0}}$)sin2($\displaystyle \theta$)e$\scriptstyle \pm$2i$\scriptstyle \phi$ = R32 . Y2$\scriptstyle \pm$2


Alexander Wagner
2000-04-14