PHY4604 R. D. Field Department of Physics Chapter4_26.doc University of Florida Euler Angles and Parity Notation:Let J2|jm> = j(j+1)|jm> D(j)(R) = general rotation operator (2j+1 dimensions) R = rotation parameters For example, opzzopyyopxxJiJiJieeeRD)()()()1()(φφφ−−−=and R = (φx, φy, φz), where hrr/)()(opopLJ=. Euler Angles:Another choice of rotation parameters are α, β, γwhere opzopyopzJiJiJieeeD)()()()1(),,(γβαγβα−−−=and 0 ≤α≤2π, 0 ≤β ≤π, 0 ≤γ≤2π. Scalars, Spinors, Vectors:A “scalar”is invariant under rotations. The operator J2is a scalar since [D(j)(R),J2]=0.A “vector”is a three component object that transforms under rotations likerRDrrr)(')1(=.A “spinor”is a two component object that transforms under rotations likesRDs~)('~)(21=.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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