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

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CHEM 356, Lecture 21, Fall 2009 1 We have previously discussed a procedure whereby the spherical harmonic functions ? ℓ,𝑚 ( 𝜃, 𝜙 ) could be constructed in terms of Θ ℓ,𝑚 ( 𝜃, 𝜙 ) and Φ 𝑚 ( 𝜙 ) functions, beginning with a knowledge of the Φ 𝑚 ( 𝜙 ) and Θ ℓ, 0 ( 𝜃 ) functions. In many of the standard quan- tum mechanics texts the spherical harmonics are expressed in terms of the Φ 𝑚 ( 𝜙 ) and Θ ℓ,𝑚 ( 𝜃 ) functions, but with the Θ ℓ,𝑚 ( 𝜃 ) functions themselves expressed in terms of associated Legendre polynomials 𝑃 𝑚 ( ? ) as Θ ℓ,𝑚 ( 𝜃 ) = [ (2 + 1)( − ∣ 𝑚 )! 2( + 𝑚 )! ] 1 2 𝑃 𝑚 ( ? ) . When this format is utilized, the spherical harmonic functions are written as ? ℓ,𝑚 ( 𝜃, 𝜙 ) = [ (2 + 1)( − ∣ 𝑚 )! 2( + 𝑚 )! ] 1 2 𝑃 𝑚 (cos 𝜃 )e 𝑖𝑚𝜙 , apart from a phase convention (which we could, as usual, express by multiplying the expression on the right–hand side by e 𝑖𝛿 ). In this case, there is a standard phase con- vention , introduced by Condon and Shortley in 1935: it is that the spherical harmonics for 𝑚 odd have the sign convention (phase factor) +1 for 𝑚 < 0, 1 for 𝑚 > 0, and that all spherical harmonics having even values of 𝑚 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 ? ℓ, 0 ( 𝜃, 𝜙 ) spherical harmonic functions. 𝑚 Θ ℓ,𝑚 ( 𝜃 ) cf. 0 0 + 1 2 𝑠 1 0 + 3 2 cos 𝜃 𝑝 ?
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