11_Hatom_new

Therefore 1 angular momentum quantum number azimuthal

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Unformatted text preview: and 0 ℓ is one. Therefore 1 Angular momentum quantum number (azimuthal) 8 9/30/2013 Radial Schrödinger equation becomes 1 2ℓ ℓ, Let 1 2ℓ ℓ 1 1, then 0 1 0 ℓ Solutions are Associated with Laguerre Polynomials, 1, Laguerre Polynomials 1 2 , Associated Laguerre Polynomials 4 6 , ℓ orbital ℓ 18 ℓ 9 ℓ 1 20 4 2p 2 10 2s 2 where 1s 21 6 2 3s 30 18 18 3p 31 96 24 3d Normalized Radial Wave Function (as in Textbook) 32 120 Put it all together and normalize ℓ ℓ 2 1! ℓ! ℓ 2 / / ℓ ℓ 3 2 ℓ Aside: There is another convention (like with Mathematica) for Associated Laguerre Polynomials ℓ ! ! 1 ℓ ! form on Wikipedia, Matlab, or Mathematica ℓ !! ℓ ℓ indexing is different ℓ When you see ℓ 1 instead of ℓ, then this is not your textbook’s convention. Then the Radial wavefunction is different ℓ ∝ / 2 ℓ ℓ 2 ℓ Other Associated Laguerre Polynomials in textbook 18 18 3 9 9/30/2013 Complete Wavefunction for Hydrogen Atom useful to retain the radial and spherical harmonics as separate orthonormal functions nlm (r , , ) Rnl (r )Yl m ( , ) where ℓ Radial ℓ 2 1! ℓ! 2 ℓ / / ℓ ℓ 2 ℓ eq. 6.47 Associated Laguerre 1/ 2 Spherical Y ( , ) (2l 1)(l m )! P |m| (cos ) 1 eim l ,m l 2(l m )! 2 Harmonics Associated Legendre principle quantum number azimuthal quantum number magnetic quantum number 0 1,2,3 … ℓ 1 0, 1, 2, … eq. 6.30 almost ℓ Unlike our previous model problems, this is a real system! Radial Wavefunctions of Hydrogen Atom orbital ,ℓ 1s 2s 2p 3s 3p 3d 10 9/30/2013 ℓ , orbital s p d Complete Wavefunction, 1s ,ℓ, , for Hydrogen Atom where / 2s 2p 3s 3p Also Ψ ,ℓ, 3d / where Hartrees 11 9/30/2013 H Atom Energy 1 1 2 1 8 1 , Hartree where 8 in atomic units • 1 in atomic units...
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This document was uploaded on 01/19/2014.

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