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# p828ps1 - Physics 828 Problem Set I Due Wednesday January...

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Physics 828: Problem Set I Due Wednesday, January 16 at 11:59 PM Note: all problems are worth 10 points unless otherwise stated 1. (20 pts.) In class, I discussed the translation operator, and explained how it can be expressed in terms of the momentum operator p . In this problem, you will consider rotation operators, and show that they can be expressed in terms of components of the angular momentum operator L . Start with a wave function ψ expressed in spherical coordinates as ψ ( r,θ,φ ). We now consider a rotation about the z axis by an angle φ 0 . Let R z ( φ 0 ) be the operator which has the following effect on the wave function ψ : R z ( φ 0 ) ψ ( r,θ,φ ) = ψ ( r,θ,φ φ 0 ) . (1) (a). By expressing ψ ( r,θ,φ φ 0 ) in a Taylor series about ψ ( r,θ,φ ), show that ψ ( r,θ,φ φ 0 ) = exp parenleftBigg φ 0 ∂φ parenrightBigg ψ ( r,θ,φ ) , (2) and hence that R z ( φ 0 ) = exp parenleftBigg φ 0 ∂φ parenrightBigg . (3) Hence, show that R z ( φ 0 ) = exp( 0 L z / ¯ h ) (4) where L z = i ¯ h ( ∂/∂φ

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p828ps1 - Physics 828 Problem Set I Due Wednesday January...

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