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Chapter4_26

# Chapter4_26 - PHY4604 R D Field Euler Angles and Parity...

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PHY4604 R. D. Field Department of Physics Chapter4_26.doc University of Florida Euler Angles and Parity Notation: Let J 2 |jm> = j(j+1)|jm> D (j) (R) = general rotation operator (2j+1 dimensions) R = rotation parameters For example, op z z op y y op x x J i J i J i e e e R D ) ( ) ( ) ( ) 1 ( ) ( φ φ φ = and R = ( φ x , φ y , φ z ) , where h r r / ) ( ) ( op op L J = . Euler Angles: Another choice of rotation parameters are α , β , γ where op z op y op z J i J i J i e e e D ) ( ) ( ) ( ) 1 ( ) , , ( γ β α γ β α = and 0 α 2 π , 0 β π , 0 γ 2 π . Scalars, Spinors, Vectors: A “scalar” is invariant under rotations. The operator J 2 is a scalar since [D (j) (R),J 2 ]=0 . A “vector” is a three component object that transforms under rotations like r R D r r r ) ( ' ) 1 ( = . A “spinor” is a two component object that transforms under rotations like s R D s ~ ) ( ' ~ ) ( 2 1 = . Parity Operator: Consider two observers with coordinate systems (x,y,z) and (x', y', z') where x' = -x , y' = -y , z' = -z . The second frame is obtained from the first by an inversion through the origin . The
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