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# set6 - Set6 Distanceexpecta.onsandmul electronatoms...

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Set 6 Distance expecta.ons and mul.‐ electron atoms

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Distance expecta.on Distance of an electron from the origin For the ground state wavefunc.on (1s) We have r = R n r ( ) 0 r R n R ( ) r 2 dr R 1,0 R ( ) = 2 Z a 0 3/2 exp Zr / a 0 [ ] r 1,0 = 4 Z 3 a 0 3 r 3 exp 2 Zr / a 0 ( ) dr 0
The average distance of an electron in a hydrogen‐like atom (one electron) r 1,0 = 4 Z 3 a 0 3 r 3 exp 2 Zr / a 0 ( ) dr 0 = = 4 Z 3 a 0 3 a 0 Z 4 Zr a 0 3 exp 2 Zr / a 0 ( ) d Zr a 0 0 = 4 a 0 Z y 3 exp 2 y ( ) dy 0 = 4 a 0 Z y 3 exp α y ( ) dy 0 = 4 a 0 Z 3 α 3 exp α y ( ) dy 0 = 4 a 0 Z 3 α 3 exp α y ( ) α 0 = 4 a 0 Z 3 α 3 1 α = 24 a 0 Z 1 2 3 = 3 a 0 2 Z The larger is the charge of the nucleus (Z) the smaller is the radius.

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