l15 - The atom Spherical coordinates Radial wavefunctions n...

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Unformatted text preview: The atom Spherical coordinates Radial wavefunctions n and Spherical harmonics Selection rules Atom wavefunctions Angular momentum Filling states The atom I Lets now tackle an especially relevant problem in quantum mechanics: the solution of electron orbitals! (Chapter 8 of Serway). We expect to have some constants of motion: 1 Total kinetic energy: quantum number is n 2 Total angular momentum L : quantum number is 3 Projection of L onto one axis: quantum number is m The time-independent Schrdinger equation in multiple dimensions involves a Laplacian 2 : planckover2pi1 2 2 m 2 + U ( r ) = E Well use spherical coordinates ( r , , ) . The Laplacian or second derivative 2 in spherical coordinates ( r , , ) becomes (equivalent but different form from Serway) 2 = 1 r 2 r ( r 2 r ) + 1 r 2 bracketleftbig 1 sin ( ) ( sin ( ) ) + 1 sin 2 ( ) 2 2 bracketrightbig The atom Spherical coordinates Radial wavefunctions n and Spherical harmonics Selection rules Atom wavefunctions Angular momentum Filling states Spherical coordinates Spherical coordi- nates: = azimuthal angle = zenith angle r = radius from origin x y z r ( r , , ) The atom Spherical coordinates Radial wavefunctions n and Spherical harmonics Selection rules Atom wavefunctions Angular momentum Filling states The atom II Assume separable variables, as we did with the 2D infinite well (Serway Eq. 8.11): ( r , , ) = R ( r )( )( ) The Schrdinger equation then becomes planckover2pi1 2 2 m braceleftBig r 2 r ( r 2 R r ) + R r 2 bracketleftbig sin ( ) ( sin ( ) ) + sin 2 ( ) 2 2 bracketrightbig bracerightBig + U ( r ) R = ER Multiply through by 2 m planckover2pi1 2 r 2 sin 2 ( ) R and rearrange. The atom Spherical coordinates Radial wavefunctions n and Spherical harmonics Selection rules Atom wavefunctions Angular momentum Filling states The atom III Multiplying through by 2 m planckover2pi1 2 r 2 sin 2 ( ) R and rearranging gives (see Serway Eq. 8.12) 1 d 2 d 2 = 2 m planckover2pi1 2 r 2 sin 2 ( )( U ( r ) E ) sin 2 ( ) R d dr ( r 2 dR dr ) sin ( ) d d ( sin ( ) d d ) . Look at this result: the left hand side depends on ( ) , while the right hand side does not. This must be true for any ( r , ) so the left hand side equals a constant! The atom Spherical coordinates Radial wavefunctions n and Spherical harmonics Selection rules Atom wavefunctions Angular momentum Filling states The atom IV Again, we have 1 d 2 d 2 = constant or d 2 d 2 ( constant ) = (1) Lets assume this is associated with the angular momentum projected on the z axis of L z involving the quantum number m . In this case we can assume solutions to (see Serway Eq. 8.13) d 2...
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This note was uploaded on 05/28/2011 for the course PHY 251 taught by Professor Rijssenbeek during the Fall '01 term at SUNY Stony Brook.

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l15 - The atom Spherical coordinates Radial wavefunctions n...

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