This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 timeindependent 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...
View
Full
Document
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.
 Fall '01
 Rijssenbeek
 Physics, Momentum

Click to edit the document details