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3502_15_octo14_LG.3

3502_15_octo14_LG.3 - 15-5 Note that by adopting a.u for...

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15-5 Note that by adopting a.u. for our unit system in eq. 15-5, it becomes implicit that we measure r in the atomic unit of distance, the bohr, which is 0.529 Å. An example of how quickly a.u. simplify expressions is that the energy of the hydrogenic atom (eq. 14-34) is given in these units as E n = Z 2 /2 n 2 . So, a ground state ( n = 1) hydrogen atom has energy 0.5 E h . So, now if we want to normalize Ψ 100 , we must solve 1 = ! 100 r , " , # ( ) ! 100 r , " , # ( ) = a 0 2 4 $ e % Zr / n e % Zr / n r 2 dr sin " d " d # 0 2 $ & 0 $ & 0 & = a 0 2 2 e % 2 Zr / n r 2 dr sin " d " 0 $ & 0 & = a 0 2 r 2 e % 2 Zr / n dr 0 & = a 0 2 n 3 4 Z 3 (15-6) or a 0 = 2 Z n ! " # $ 3/ 2 (15-7) If we use this in eq. 15-5 (now explicitly taking n = 1; we've carried it as a variable up until now because that will help to explain later results) we have ! 100 r , " , # ( ) = Z 3 /2 $ e % Zr (15-8) This is really a rather simple form. We can carry out similar manipulations for each additional choice of
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