11_Hatom_new

The radial value is fixed at the max of the radial

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Unformatted text preview: 4d 4f Bohr didn’t quite get it right. 1,2,3, … all states had angular momentum Bohr Schrödinger ℓℓ 1 ℓ 0,1,2, … 1 ℓ can equal zero 13 9/30/2013 You might think that this implies that 1s electron is most likely found at nucleus Ground State Wavefunction 1 100 1s a 3 0 e r / a0 but this is three dimensional and the volume of the radial shell at 0 is going to zero ∗ e 2 r / a 0 1*s 1s 12s 3 a0 → integrate the probability density over and , but not Radial Distribution Function ∗ ,, ,, volume element ∗ ∗ , , is orthonormal ∗ ∗ Radial Distribution Function Volume of Shell 4 3 4 3 4 3 4 3 4 4 3 4 3 2 2 3 2 3 ignoring higher order terms ∗ Radial Distribution Function ∗ ∗ weighting proportional to shell volume → → 14 9/30/2013 Radial Distribution Function ∗ / 2 / , / 4 4 2 / 2 / / 2 Radial Ground State Wavefunction substitute / Find Most Probable Value of 4 / 2 / / / 2 / 2 0 rearranging rhs just like Bohr Theory Expectation Value of Ground State Radial Position ** r1s 1*s r 1s d R10Y00 rR10Y00 d 2 0 0 0 ** R10Y00 rR10Y00 r 2 sin dddr 0 0 2 r 3 R10 dr 2 r 3 R10 dr 0 use 0 2 0 2 0 2 Y00 sin dd instead of volume element separate angular and radial coord. 1 1 2 2 r R dr sin d r 3 R10 dr 2 r 3 R10 dr 0 0 0 0 2 2 2 3 4r 1 r / a0 4 3! 3 2 r / a0 3 r 2 3 e dr 3 a0 dr 0 3 e 4 0 a0 a0 2 / a0 2 a0 3 2 10 rmost probable raverage since ! ∗ 1, Can take but lets check 1 sin dd 4 15 9/30/2013 Radial Distribution Functions of Orbitals Radial Expectation Values 0.75 Ȧ 6 3.12 Ȧ 27 2 Radial Expectation Values Radial Distribution Functions (Blue) 3s rnl 1 a0 3n 2 l (l 1) 2 0.1 R rri n l 2 13.5 R rri n l 2 rri 2 0.05 _ 0 0 5 10 15 20 25 rri 0.1 R rri n l 2 3p 12.5 R rri n l 2 rri 2 0.05 _ 0 0 5 10 15 20 25 rri 3d 0.1 R rri n...
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