Lecture21 - CHEM 356, Lecture 21, Fall 2009 1 We have...

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Unformatted text preview: CHEM 356, Lecture 21, Fall 2009 1 We have previously discussed a procedure whereby the spherical harmonic functions ? , ( , ) could be constructed in terms of , ( , ) and ( ) functions, beginning with a knowledge of the ( ) and , ( ) functions. In many of the standard quan- tum mechanics texts the spherical harmonics are expressed in terms of the ( ) and , ( ) functions, but with the , ( ) functions themselves expressed in terms of associated Legendre polynomials ( ? ) as , ( ) = [ (2 + 1)( )! 2( + )! ] 1 2 ( ? ) . When this format is utilized, the spherical harmonic functions are written as ? , ( , ) = [ (2 + 1)( )! 2( + )! ] 1 2 (cos )e , apart from a phase convention (which we could, as usual, express by multiplying the expression on the righthand side by e ). In this case, there is a standard phase con- vention , introduced by Condon and Shortley in 1935: it is that the spherical harmonics for odd have the sign convention (phase factor) +1 for < 0, 1 for > 0, and that all spherical harmonics having even values of have a phase factor +1. Notice that this phase convention corresponds precisely to what we obtained using ladder operators to construct the more general spherical harmonics from the simpler set of ?...
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This note was uploaded on 02/28/2011 for the course CHEM 356 taught by Professor Prof.iaskjd during the Fall '09 term at Waterloo.

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Lecture21 - CHEM 356, Lecture 21, Fall 2009 1 We have...

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