Spherical harmonics

Spherical harmonics - And even more surprising, (11.31)...

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Spherical harmonics The key issue about three-dimensional motion in a spherical potential is angular momentum. This is true classically as well as in quantum theories. The angular momentum in classical mechanics is defined as the vector (outer) product of and , (11.26) This has an easy quantum analog that can be written as (11.27) After exapnsion we find (11.28) This operator has some very interesting properties: (11.29) Thus (11.30)
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Unformatted text preview: And even more surprising, (11.31) Thus the different components of are not compatible (i.e., can't be determined at the same time). Since commutes with we can diagonalise one of the components of at the same time as . Actually, we diagonalsie , and at the same time! The solutions to the equation (11.32) are called the spherical harmonics. heck that is independent of ! The label corresponds to the operator , (11.33)...
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This note was uploaded on 11/09/2011 for the course PHY PHY2053 taught by Professor Davidjudd during the Fall '10 term at Broward College.

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Spherical harmonics - And even more surprising, (11.31)...

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