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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 11 1. [20 pts] In order to derive the properties of the angular momentum eigenstate wavefunctions, we need to determine the action of the angular momentum operator in spherical coordinates. Just as we have ( x | P x | ) = i planckover2pi1 d dx ( x | ) , we should find a similar expression for ( r | vector L | ) . From vector L = vector R vector P and our knowledge of momentum operators, it follows that ( r | vector L | ) = planckover2pi1 parenleftbigg vectore x parenleftbigg y d dz z d dy parenrightbigg + vectore y parenleftbigg z d dx x d dz parenrightbigg + vectore z parenleftbigg x d dy y d dx parenrightbiggparenrightbigg ( r | ) . The coordinates are defined via the transformations x = r sin cos y = r sin sin z = r cos and the inverse transformations r = radicalbig x 2 + y 2 + z 2 = arctan( radicalbig x 2 + y 2 z ) = arctan( y x ) , while their derivatives can be related via expansions such as d dx = r x r + x...
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