851HW11_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 11 Topics Covered: Orbital angular momentum, center-of-mass coordinates Some Key Concepts: angular degrees of freedom, spherical harmonics 1. [20 pts] In order to derive the properties of the spherical harmonics, we need to determine the action of the angular momentum operator in spherical coordinates. Just as we have ( x | P x | ψ ) = − i planckover2pi1 d dx ( x | ψ ) , we should find a similar expression for ( rθφ | vector L | ψ ) . From vector L = vector R × vector P and our knowledge of momentum operators, it follows that ( rθφ | vector L | ψ ) = − ı 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 ( rθφ | ψ ) . Cartesian coordinates are related to spherical coordinates via the transformations x = r sin θ cos φ y = r sin θ sin φ z = r cos θ and the inverse transformations r = radicalbig x 2 + y 2 + z 2 θ = arctan( radicalbig x 2 + y 2 z ) φ = arctan( y x ) ....
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851HW11_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

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