Chapter4_20 - PHY4604 R D Field Rotations in Spin-Space...

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PHY4604 R. D. Field Department of Physics Chapter4_20.doc University of Florida Rotations in Spin-Space Taylor Series Expansion: The Taylor series expansion for the function f(x) is given by L + + + + = = + = 3 3 3 2 2 2 0 ) ( ! 3 ) ( ! 2 ) ( ) ( ) ( ! ) ( dx x df a dx x df a dx x df a x f dx x f d n a a x f n n n n but = = = = 0 / ! ) ( n n n n dx d a iap dx d n a e e a U x h and ) ( ) ( ) ( ) ( / x f e x f a U a x f x iap h = = + . For a = ε << 1 then dx x df x f x f p i x f e x f x p i x ) ( ) ( ) ( ) / 1 ( ) ( ) ( / ε ε ε ε + = + = = + h h The operator h / x p is the generator of spatial translations. The Taylor series expansion for the function f( φ ) is given by = = + 0 ! ) ( n n n n f n f φ ϕ ϕ φ and = = = = 0 / ! ) ( n n n n iL z n e e R z φ ϕ ϕ φ ϕ ϕ h and ) ( ) ( ) ( ) ( / φ φ ϕ ϕ φ ϕ f e f R f z iL z h = = + , where I used φ = h i L z . The operator h / z L is the generator of rotations about the z-axis. Rotations in Spin-Space: For spin ½ spinors we have z z S σ 2 h = > >= >= >= >= χ χ χ ϕ ϕ χ χ ϕ σ ϕ | | | ) ( ) ( | ' | 2 / / z z i iS z e e R h and ) sin( ) cos( ) ( ! 1 ) ( ! 1 ) ( 2 1 2 1 2 2 2 1 k k k n odd n k i k n even n k i i k k i n n e R k k ϕ σ ϕ ϕ σ ϕ ϕ σ ϕ + = + = = (k = 1, 2, 3) +
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