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Chapter4_25 - ∂ ∂ − ∂ ∂ − = x y y x i L op z h...

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PHY4604 R. D. Field Department of Physics Chapter4_25.doc University of Florida Rotations in Quantum Mechanics In quantum mechanics we have unitary operators that rotate the wavefunction ) ( r r Ψ to a new wavefunction ) ( ) ( ' r R U v Ψ = Ψ “Active View”: Think of the physical system being rotated. Hold the coordinate system fixed and rotate the physical system. “Passive View”: Keep the physical system fixed and rotate the coordinates by R -1 . Of course these two views are equivalent and ) ( ) ( ) ( ) ( ' 1 r R r R U r v v r Ψ = Ψ = Ψ or ) ( ) ( ) ( 1 r R r R U v v Ψ = Ψ The unitary operators U(R) form a group with exactly the same properties as the group R( φ x , φ y , φ z ) . Infinitesimal Rotations About the z-Axis: For infinitesimal rotations about the z-axis ) , , ( ) ( ) , , ( ) ( 1 z x y y x r R z y x R U z z z z δφ + Ψ = Ψ = Ψ v since + = = = = z x y x z y x r I I r R r R z z z z z z z z z z 1 0 0 0 1 0 1 ) ( ) ( ) ( 1 r r r . Thus L h + Ψ Ψ = + Ψ + Ψ = + Ψ = Ψ ) , , ( ) ( ) , , ( ) ( ) , , ( ) , , ( ) , , ( ) , , ( ) ( 2 z y x L z y x O z y x y x x y z y x z x y y x z y x R U op z z i z z z z z where
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Unformatted text preview: ∂ ∂ − ∂ ∂ − = x y y x i L op z h ) ( and where I used L + ∂ Ψ ∂ + ∂ Ψ ∂ + Ψ = ∂ + Ψ ∂ + + Ψ = + + Ψ x z y x x z y x z y x x z y x z y x z y x y x y x y y x ) , , ( ) , , ( ) , , ( ) , , ( ) , , ( ) , , ( ε with ε x = δφ z y and ε y = -δφ z x. Thus, (L z ) op is the generator of rotations about the z-axis and op z z i L z e R U ) ( ) ( φ h − = and op z z i op y y i op x x i L L L e e e R U ) ( ) ( ) ( ) ( h h h − − − = . The rotation operator U(R) is unitary since the generators (L i ) op are hermitian. x-axis φ z physical system Ψ physical system Ψ ' y-axis x-axis φ z physical system Ψ ' y-axis x'-axis y'-axis...
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